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\begin_layout Title
Language Learning Diary - Part Five
\end_layout
\begin_layout Date
Jan 2022 - March 2022
\end_layout
\begin_layout Author
Linas Vepštas
\end_layout
\begin_layout Abstract
The language-learning effort involves research and software development
to implement the ideas concerning unsupervised learning of grammar, syntax
and semantics from corpora.
This document contains supplementary notes and a loosely-organized semi-chronol
ogical diary of results.
The notes here might not always makes sense; they are a short-hand for
my own benefit, rather than aimed at you, dear reader!
\end_layout
\begin_layout Section*
Introduction
\end_layout
\begin_layout Standard
Part Five of the diary on the language-learning effort continues work on
the English dataset.
\end_layout
\begin_layout Standard
Good progress was made.
It appears that we have a high-quality, stable, debugged clustering algorithm
in place, and it is generating good-quality clusters.
However, dozens of questions and hypothesis arise, which is a bit overwhelming.
The lack of any kind of theoretical foundations for any of this stuff is
frustrating.
Without theory, it is hard to get strong insight.
\end_layout
\begin_layout Standard
At any rate, results seem solid enough that we can now move on to the next
steps of the program.
I'm hoping to start work on multi-sentence and cross-paragraph correlations
in the coming months.
Although, before doing this, perhaps a comparison to the hand-built LG
grammars is in order.
\end_layout
\begin_layout Section*
Summary Conclusions
\end_layout
\begin_layout Standard
A summary of what is found in this part of the diary:
\end_layout
\begin_layout Itemize
From Experiment-10: Ranked-MI (Variational MI) seems to mostly make good
cluster recommendations, except when it doesn't.
Since the actual merge proceeds via a Jaccard-type voting in of disjuncts,
it is possible that the poor recommendations results in few contributions.
But there is no obvious way to automate a quality measurement.
\end_layout
\begin_layout Itemize
From Experiment-10: A list of the first 31 clusters is presented.
It seems OK but not great.
Hard to say, as the Jaccard-selection mechanism selects membership, and
we don't have a window into that.
Is there a way of visualizing the
\begin_inset Quotes eld
\end_inset
word-sense
\begin_inset Quotes erd
\end_inset
of a cluster?
\end_layout
\begin_layout Itemize
From Experiment-10: A fairly large number of merges are expanding or merging
prior classes.
This suggests that
\begin_inset Quotes eld
\end_inset
preferential attachment
\begin_inset Quotes erd
\end_inset
is hard at work, and so we should not be surprised to later observe scale-free
results.
\end_layout
\begin_layout Itemize
This implies that ranked-MI/variational-MI automatically, inherently makes
these kind of preferential-attachment suggestions.
How does this work? What are the details? Are there alternatives?
\end_layout
\begin_layout Itemize
As clustering proceeds, there are several dozen different measurements and
indicator values that can be tracked, including the dataset entropy, the
MMT-Q value, the size of clusters, the ranked-MI of the top-ranked word-pair,
and so on.
Many graphs of these indicators are presented.
Experiment-10 tracks four of these; experiments 11 and 12 tracks dozens
of them.
A significant portion of the parameter space is explored.
\end_layout
\begin_layout Itemize
Graphs of the word-disjunct marginal entropy distribution is presented.
This for the dataset, before clustering is started.
It appears to be a normal (Gaussian) distribution.
\end_layout
\begin_layout Itemize
Multiple alternatives to the variational-MI (ranked-MI) are proposed and
developed.
These are computed for the top-200 most common words in the dataset (
\emph on
i.e.
\emph default
for
\begin_inset Formula $N\left(N+1\right)/2$
\end_inset
word-pairs, for
\begin_inset Formula $N=200$
\end_inset
.) Although they make different ranking recommendations, the overall distribution
s are nearly identical.
Although roughly Gaussian, these have a very fat tail to the left (
\emph on
i.e.
\emph default
on the negative side: they find a lot of dis-similarity.) The fat tail is
hypothesized to be driven by the number of distinct word-senses (
\emph on
i.e.
\emph default
word-senses should be dis-similar.) All of these alternatives are more computati
onally intensive than the ranked-MI, and thus are not really practical to
apply.
It is hypothesized that all of these might still end up selecting the same
in-group (
\emph on
i.e.
\emph default
although the top-ranked pairs differ, the in-groups would not.)
\end_layout
\begin_layout Itemize
From Experiment-11: The top ranked-MI pair is used to form the initial seed
of a cluster, and then the regular MI is used to nominate additional members.
When the Jaccard overlap of the nominees is computed, it is surprisingly
low.
Why? Is it because MI is pair-wise, and Jaccard is group-wise?
\end_layout
\begin_layout Itemize
Despite the low overlap, the majority of sections get merged, only because
their counts are below the noise-floor.
How do low-count disjuncts influence the MI?
\end_layout
\begin_layout Itemize
Setting the noise-floor causes disjuncts with a count below this floor to
be swept up into the group.
Later, during the preferential attachment and growth of groups, is it possible
that this
\begin_inset Quotes eld
\end_inset
random noise
\begin_inset Quotes erd
\end_inset
hijacks the initial cluster, twisting it in a different direction from
where it started? Is it causing the signal to be washed out? What happens
if we set the noise-floor to zero?
\end_layout
\begin_layout Itemize
After running the algo for about 1060 steps, about 530 word-classes were
created (the fact that its about a half is unexplained, but gives a hint
of the role of preferential attachment).
When comparing the
\begin_inset Formula $N\left(N-1\right)/2$
\end_inset
class-pairs (for
\begin_inset Formula $N=530)$
\end_inset
, about 93% of them have a similarity of zero: that is, they are completely
orthogonal.
That's a good thing: we wanted to have classes that are distinct from one-anoth
er, and that is what we are getting.
Of the remaining 7%, the distribution of MI is almost perfectly a Gaussian,
centered at about MI of negative 3, and an stddev of 3.5.
Negative MI is also good: it means that even if they are not perfectly
orthogonal, they are almost-so.
\end_layout
\begin_layout Itemize
The above stands in contrast to the word-word orthogonality of the remaining
(as yet unmerged) words: more than 80% of them have a non-zero similarity.
The distribution of these is almost Gaussian, with a bit of a fat tail
towards the low-end.
Compared to the pre-merge MI distribution, there is little change, except
perhaps in the fat tail.
\end_layout
\begin_layout Standard
That pretty much wraps things up for this chapter.
Merge experiments are ongoing, and further results will be presented in
later diary chapters.
I don't expect many changes from the above; just some longer runs with
maybe clearer results.
A noise=0 run is clearly called for.
Examining orthogonality as a function of time is also interesting, to be
reported later.
\end_layout
\begin_layout Section*
TODO
\end_layout
\begin_layout Standard
A list of things to do, here or later:
\end_layout
\begin_layout Itemize
In population genetics, a neutral evolutionary model with a static population
size results in a power-law distribution of alleles in the population.
Can this process model also explain the power-law (Zipfian, square-root-Zipfian
) distributions seen elsewhere? There is a similar set of results in ecology,
with regards to species distribution in an ecological niche.
\end_layout
\begin_layout Section*
Expt-10 Merge exploration (Jan 2022)
\end_layout
\begin_layout Standard
The end of Diary Part Four describes a dataset with several thousand merged
words in it.
However, that dataset had assorted issues, and recreating it before investing
time in characterizing it seems like a good idea.
There were numerous (and ongoing) bug-fixes made.
Phew.
5 Jan 2022 Looks like the last of the bugs are now fixed.
This took two weeks of very tedious debugging.
Ouch.
Ready to restart, at last.
\end_layout
\begin_layout Standard
Now that we are ready, the questions below arise.
\end_layout
\begin_layout Subsection*
Things worth exploring.
\end_layout
\begin_layout Standard
What have we actually got? Some questions about the dataset:
\end_layout
\begin_layout Itemize
Distribution: size of word-class vs.
rank.
This was previously examined in
\begin_inset Quotes eld
\end_inset
diary part one
\begin_inset Quotes erd
\end_inset
, page 99, for a different collection of merge algorithms (and earlier,
different datasets).
Do we still get something similar?
\end_layout
\begin_layout Itemize
Above,
\begin_inset Quotes eld
\end_inset
size
\begin_inset Quotes erd
\end_inset
might mean
\begin_inset Quotes eld
\end_inset
number of words in the word-class
\begin_inset Quotes erd
\end_inset
or it might mean
\begin_inset Quotes eld
\end_inset
number of disjuncts in word-class
\begin_inset Quotes erd
\end_inset
or it might mean
\begin_inset Quotes eld
\end_inset
number of disjuncts in word-class with count weighting
\begin_inset Quotes erd
\end_inset
.
Oof.
It would be nice to have a dashboard for this, instead of lots of manual
work.
\end_layout
\begin_layout Itemize
Distribution of word-senses.
That is, how many words participate in more than one word-class? If they
do participate in more than one word-class, what is the weight in each
class?
\end_layout
\begin_layout Itemize
For words that belong to word-classes, what fraction of their weight remains
unassigned to any word-class?
\end_layout
\begin_layout Itemize
Distribution of MI of pairs of word-classes.
We might hope that this is low, so that different word-classes are different
from one another.
\end_layout
\begin_layout Itemize
Distribution of self-MI of word-classes.
One might hope that this is high, so that the word-classes do not share
much in common with other words or word-classes.
\end_layout
\begin_layout Itemize
As above, but distribution of MI of pairs consisting of a word-class, and
a word.
\end_layout
\begin_layout Itemize
Prior to starting the merge, there's an MI between words and disjuncts.
I don't recall examining that in detail, before.
Then, after the merge, how does this change?
\end_layout
\begin_layout Standard
That's a lot of questions.
Not clear which ones should be answered first.
\end_layout
\begin_layout Subsection*
Round 43
\end_layout
\begin_layout Standard
Done with bug-fixing.
\end_layout
\begin_layout Standard
I'm currently using `
\family sans
r9-sim-200.rdb
\family default
` for marginals plus the similarities for the top 200 most frequent words.
Then run
\begin_inset listings
inline false
status open
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cp -pr r9-sim-200.rdb r10-merge.rdb
\end_layout
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guile -l cogserver-gram.scm
\end_layout
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(in-group-cluster covr-obj 0.5 0.2 4 200 100)
\end_layout
\end_inset
\end_layout
\begin_layout Standard
This works great until merge 43 where we get
\end_layout
\begin_layout Standard
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------ Round 43 Next in line: ranked-MI = 5.6267 MI = 5.1464 (`could must
would should may will can might`, `to`)
\end_layout
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In-group size=5: `to` `could must would should may will can might` `you`
`I` `we`
\end_layout
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\end_layout
\begin_layout Standard
So that looks like a bug.
Issues:
\end_layout
\begin_layout Itemize
Hard to believe that this is the top ranked-MI pair.
\end_layout
\begin_deeper
\begin_layout Itemize
Is the MI being computed correctly? It almost surely is, but still ...
??
\end_layout
\begin_layout Itemize
Should the ranked-MI for a cluster be de-rated, say, by the number of words
in the cluster?
\end_layout
\begin_layout Itemize
The next 10 highest ranked-MI merges look great! So maybe somehow the ranked
MI for clusters is wrong?
\end_layout
\begin_layout Itemize
Since merging is via a Jaccard-overlap selection mechanism, perhaps most
of the contributing disjuncts in the bad recommendations will be ignored?
\end_layout
\end_deeper
\begin_layout Standard
\begin_inset Separator plain
\end_inset
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\begin_layout Subsection*
Round 88 - Majority voting bug?
\end_layout
\begin_layout Standard
Subsequent merges look ..
pretty good, except when they don't, and then they look ugly.
\end_layout
\begin_layout Standard
At round 88 it goes nuts: it merges a few disjuncts of
\begin_inset Quotes eld
\end_inset
As as
\begin_inset Quotes erd
\end_inset
together.
Then it merges zero of them into
\begin_inset Quotes eld
\end_inset
As as.i
\begin_inset Quotes erd
\end_inset
, and then hits an inf loop, because what is left after the merge still
has a high MI.
\end_layout
\begin_layout Standard
Conclude: the majority-voting scheme left behind (left unmerged) too much;
enough that the MI between what is left is still high.
Can we redefine the voting procedure to lessen this?
\end_layout
\begin_layout Subsection*
Bug Fixes
\end_layout
\begin_layout Standard
There were multiple bugs, difficult to locate.
Some cross-section merges were being done wrong.
A bug in cog-delete! was erasing data in DB, thus restarts loaded missing/corru
pt data.
As of 15 Jan all seems to be fixed.
\end_layout
\begin_layout Subsection*
Round 31
\end_layout
\begin_layout Standard
After fixes, the first 31 rounds merge the following words.
The is with quorum=0.5 commonality=0.2 noise=4.
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Expansion of group above!
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his the a he I
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odd but plausible
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Yuck! Merge N=4,7,1
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</row>
<row>
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13
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
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\end_layout
\end_inset
</cell>
</row>
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
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\end_layout
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</cell>
</row>
<row endhead="true">
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
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\end_layout
\end_inset
</cell>
</row>
<row>
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\begin_inset Text
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16
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</cell>
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
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\end_inset
</cell>
</row>
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\begin_inset Text
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17
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
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\end_inset
</cell>
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\begin_inset Text
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\end_layout
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
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\end_inset
</cell>
</row>
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\begin_inset Text
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
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\begin_inset Text
\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
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\end_layout
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
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\end_inset
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
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\end_layout
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</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
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</cell>
</row>
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\begin_inset Text
\begin_layout Plain Layout
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<cell alignment="center" valignment="top" leftline="true" usebox="none">
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24
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
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\end_layout
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</cell>
</row>
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\begin_inset Text
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
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</row>
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26
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
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</cell>
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<row>
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\begin_inset Text
\begin_layout Plain Layout
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28
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</cell>
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</cell>
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29
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</cell>
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</cell>
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<row>
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30
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</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\end_layout
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</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Meh.
Expand 11
\end_layout
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</cell>
</row>
<row>
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\begin_inset Text
\begin_layout Plain Layout
31
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</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
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on upon
\end_layout
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</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
great!
\end_layout
\end_inset
</cell>
</row>
</lyxtabular>
\end_inset
\end_layout
\begin_layout Standard
\begin_inset VSpace defskip
\end_inset
\end_layout
\begin_layout Standard
Notable is that a fairly large number of merges are expanding or merging
prior classes.
This is curious.
This suggests that
\begin_inset Quotes eld
\end_inset
preferential attachment
\begin_inset Quotes erd
\end_inset
is hard at work, and so we should not be surprised to later observe scale-free
results.
\end_layout
\begin_layout Standard
This implies that ranked-MI/variational-MI automatically, inherently makes
these kind of preferential-attachment suggestions.
How does this work? What are the details? Are there alternatives?
\end_layout
\begin_layout Standard
Regarding the mediocre merge suggestions: The Jaccard-selection mechanism
selects membership, and we don't have a window into that.
Its possible that the mediocre merge recommendations work out so that very
few disjuncts get merged into the final cluster.
Is there a way of visualizing the
\begin_inset Quotes eld
\end_inset
word-sense
\begin_inset Quotes erd
\end_inset
of a cluster?
\end_layout
\begin_layout Standard
Also being tracked, per merge: rows, cols, lcnt, rcnt, size, sparsity, entropy,
ranked-mi, mmt-q.
These look good, up to round 28, when lcnt=0, size=0 and some divide-by-zero
garbage up the other stats! After that point, the lcnt off by a little
bit.
\end_layout
\begin_layout Standard
At the conclusion of round 27, mmt-q=+inf.0 is already wrong.
But this value gets recorded as part of round 28.
During extension of round 27, some occasional garbage:
\end_layout
\begin_layout Standard
\begin_inset listings
inline false
status open
\begin_layout Plain Layout
MI(`in`, `into`) = +inf.0 rank-MI = +nan.0
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Turns out is was a bad wild-card de-reference that clobbered the wildcard
anchor.
\end_layout
\begin_layout Subsection*
Merge Datasets
\end_layout
\begin_layout Standard
The following merge parameters are being pursued in the following datasets:
\begin_inset VSpace defskip
\end_inset
\end_layout
\begin_layout Standard
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="5" columns="4">
<features tabularvalignment="middle">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
name
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
quorum
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
commonality
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
noise
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
r10-mrg-q0.5-c0.2.rdb
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
0.5
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
0.2
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
4
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
r10-mrg-q0.6-c0.3.rdb
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
0.6
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
0.3
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
4
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
r10-mrg-q0.7-c0.4.rdb
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
0.7
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
0.4
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
4
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
r10-mrg-q0.8-c0.5.rdb
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
0.8
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
0.5
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
4
\end_layout
\end_inset
</cell>
</row>
</lyxtabular>
\end_inset
\end_layout
\begin_layout Standard
\begin_inset VSpace defskip
\end_inset
\end_layout
\begin_layout Standard
Spot sampling of the merges looks much like the above: some pretty good-looking
merges interspersed with some head-scratchers.
This occasional ugliness continues deeply into the merge stream.
What is not clear is how many disjuncts are involved in the uglies, and,
if they occur early, exactly how much
\begin_inset Quotes eld
\end_inset
noise
\begin_inset Quotes erd
\end_inset
they carry away.
\end_layout
\begin_layout Standard
What does come clear from the spot inspection is that ranked-MI is starting
to make some pretty poor recommendations.
Basically, the clusters get an
\begin_inset Quotes eld
\end_inset
unnaturally
\begin_inset Quotes erd
\end_inset
large ranked-MI, because they are so large, even though the raw MI to what
is being suggested is relatively low.
That is, the use of ranked-MI is the reason why the clusters grow preferentiall
y.
\end_layout
\begin_layout Standard
For this round, four stats are tracked: the dataset sparsity, the dataset
entropy, the ranked MI of the next pair to be selected, and the
\begin_inset Formula $MM^{T}Q$
\end_inset
as defined in the previous diary.
\end_layout
\begin_layout Standard
Here's all four of these on one graph:
\end_layout
\begin_layout Standard
\align center
\begin_inset Graphics
filename p5-en-merge/log-0.5-0.2.eps
width 80text%
\end_inset
\end_layout
\begin_layout Standard
Here's just sparsity, for all four merge parameters.
Although it is increasing, it is increasing by very small amounts.
The quorum=0.8 run crashed in a way that doesn't make it worth restarting
(see newer data below).
\end_layout
\begin_layout Standard
\align center
\begin_inset Graphics
filename p5-en-merge/sparsity.eps
width 80text%
\end_inset
\end_layout
\begin_layout Standard
The dataset total MM^T entropy.
It is dropping because the merges are
\begin_inset Quotes eld
\end_inset
bringing order to the chaos
\begin_inset Quotes erd
\end_inset
.
\end_layout
\begin_layout Standard
\align center
\begin_inset Graphics
filename p5-en-merge/entropy.eps
width 80text%
\end_inset
\end_layout
\begin_layout Standard
The ranked-MI of the next pair to merge drops, because, naturally, the highest-r
anked pairs are merged.
It spiked whenever a large cluster is created: the shear size of this cluster
guarantees that it will have a large ranked-MI, irrespective of the absolute-MI.
\end_layout
\begin_layout Standard
\align center
\begin_inset Graphics
filename p5-en-merge/ranked-mi.eps
width 80text%
\end_inset
\end_layout
\begin_layout Standard
The value
\begin_inset Formula $Q$
\end_inset
is being used as a
\begin_inset Quotes eld
\end_inset
constant
\begin_inset Quotes erd
\end_inset
to offset the common-MI to a non-negative range.
The definition was the log of the sum-squared marginals:
\begin_inset Formula
\[
Q=-\log_{2}\sum_{d}P\left(*,d\right)P\left(*,d\right)
\]
\end_inset
This is
\begin_inset Quotes eld
\end_inset
constant
\begin_inset Quotes erd
\end_inset
for a fixed dataset, in that it is independent of the word and disjunct.
As merges proceed, it changes.
\end_layout
\begin_layout Standard
\align center
\begin_inset Graphics
filename p5-en-merge/mmtq.eps
width 80text%
\end_inset
\end_layout
\begin_layout Standard
There seems to be a big changes in the quorum=0.7 curve, just before the
800'th merge.
Unfortunately, this code failed to track the merged words, so unclear what
that's about.
\end_layout
\begin_layout Standard
Other issues: the above run fails to compute the marginals correctly, and
thus some of the similarities could easily be off.
This particular bug is fixed, and a newer run is presented further below.
\end_layout
\begin_layout Standard
Notes:
\end_layout
\begin_layout Itemize
The quorum=0.8 run died prematurely, after performing 986 merges.
At the conclusion, it only had 502 word-classes, so clearly, many classes
are being recombined and expanded.
\end_layout
\begin_layout Itemize
Failed to write out the MemberLinks on all four datasets, and so these are
useless for creating dictionaries.
\end_layout
\begin_layout Section*
Marginal Entropy Distributions (Jan 2022)
\end_layout
\begin_layout Standard
In the paper
\begin_inset Quotes eld
\end_inset
Connector Set Distributions
\begin_inset Quotes erd
\end_inset
, an extensive examination of various relationships between word distributions,
entropy, mutual information and rank is explored.
Below, we explore a bit more in that vein, that apparently was not previously
explored.
\end_layout
\begin_layout Standard
In Diary Part Three we looked at the word-rank vs.
support (page 10).
Here, we look at rank vs.
right-entropy and rank vs.
right-MI as well as rank vs log-probability.
The goal of asking these questions is two-fold.
\end_layout
\begin_layout Itemize
The current merge algo uses ranked-MI to select the next word-pair to anchor
a merge.
Perhaps some other linear combination of MI and, say, right-entropy might
be better?
\end_layout
\begin_layout Itemize
The current merge algo uses a majority-voting scheme, by computing an overlap.
Perhaps some log-overlap, of some word-disjunct MI would be better?
\end_layout
\begin_layout Standard
Anyway, these relationships seem interesting.
\end_layout
\begin_layout Subsection*
Definitions
\end_layout
\begin_layout Standard
Some definitions are in order.
\begin_inset Foot
status collapsed
\begin_layout Plain Layout
The definitions here are the same as given in earlier texts.
The code that implements these was developed primarily in 2017, and can
be found in the `opencog/matrix` subdirectory of the git repo at
\begin_inset CommandInset href
LatexCommand href
name "AtomSpace"
target "https://github.com/opencog/atomspace/tree/master/opencog/matrix"
literal "false"
\end_inset
.
See, in particular, `frequency.scm` and `entropy.scm`.
\end_layout
\end_inset
This repeats definitions given earlier.
For a word-disjunct pair
\begin_inset Formula $\left(w,d\right)$
\end_inset
the frequentist probability is
\begin_inset Formula $P\left(w,d\right)=N\left(w,d\right)/N\left(*,*\right)$
\end_inset
.
Since it has two arguments, this can be called the 'joint probability'
or 'joint frequency'.
\end_layout
\begin_layout Subsubsection*
Total and Marginal Entropies
\end_layout
\begin_layout Standard
The total entropy is
\begin_inset Formula
\[
H=H_{\mbox{tot}}=-\sum_{w,d}P\left(w,d\right)\log_{2}P\left(w,d\right)
\]
\end_inset
The right-marginal entropy for a given word
\begin_inset Formula $w$
\end_inset
is
\begin_inset Formula
\[
h_{\mbox{right}}\left(w\right)=-\sum_{d}P\left(w,d\right)\log_{2}P\left(w,d\right)
\]
\end_inset
The fractional right-marginal entropy for a given word
\begin_inset Formula $w$
\end_inset
is
\begin_inset Formula
\[
H_{\mbox{right}}\left(w\right)=\frac{h_{\mbox{right}}\left(w\right)}{P\left(w,*\right)}
\]
\end_inset
The goal of the division by the marginal probability
\begin_inset Formula $P\left(w,*\right)$
\end_inset
of the word
\begin_inset Formula $w$
\end_inset
is to rescale the entropy into a sensible range.
For the current dataset, this is in the range of 5 to 25, as graphed below.
The
\begin_inset Formula $h_{\mbox{left}}\left(d\right)$
\end_inset
and
\begin_inset Formula $H_{\mbox{left}}\left(d\right)$
\end_inset
for disjuncts
\begin_inset Formula $d$
\end_inset
are defined likewise.
Note that
\begin_inset Formula
\[
H_{\mbox{tot}}=\sum_{w}P\left(w,*\right)H_{\mbox{right}}\left(w\right)=\sum_{d}P\left(*,d\right)H_{\mbox{left}}\left(d\right)
\]
\end_inset
can be used as a consistency check on the numerical results.
\end_layout
\begin_layout Standard
The marginal probabilities also allow two more kinds of entropies to be
defined.
These are the left and right entropies of the left and right marginal probabili
ties:
\begin_inset Formula
\[
S_{\mbox{right}}=\sum_{w}P\left(w,*\right)\log_{2}P\left(w,*\right)
\]
\end_inset
and likewise
\begin_inset Formula
\[
S_{\mbox{left}}=\sum_{d}P\left(*,d\right)\log_{2}P\left(*,d\right)
\]
\end_inset
One might argue that the names
\begin_inset Quotes eld
\end_inset
left
\begin_inset Quotes erd
\end_inset
and
\begin_inset Quotes eld
\end_inset
right
\begin_inset Quotes erd
\end_inset
are incorrectly interchanged in the above, but this is the naming convention
used in the implementation, and changing it risks introducing bugs and
confusion.
\end_layout
\begin_layout Standard
These two are related to the total mutual information (as defined below)
as
\begin_inset Formula
\[
MI_{\mbox{tot}}=S_{\mbox{right}}+S_{\mbox{left}}-H_{\mbox{tot}}
\]
\end_inset
\end_layout
\begin_layout Subsubsection*
Total and Marginal Mutual Information
\end_layout
\begin_layout Standard
The mutual information for a word-disjunct pair
\begin_inset Formula $\left(w,d\right)$
\end_inset
is
\begin_inset Formula
\[
mi\left(w,d\right)=P\left(w,d\right)\log_{2}\frac{P\left(w,d\right)}{P\left(*,d\right)P\left(w,*\right)}
\]
\end_inset
As always, the fractional MI is
\begin_inset Formula
\[
MI\left(w,d\right)=\frac{mi\left(w,d\right)}{P\left(w,d\right)}
\]
\end_inset
thus scaling it into a sensible range of -5 to +20, as shown below.
The marginals proceed likewise, as above, so that the right marginal
\begin_inset Formula $mi$
\end_inset
for word
\begin_inset Formula $w$
\end_inset
is
\begin_inset Formula
\[
mi_{\mbox{right}}\left(w\right)=mi\left(w,*\right)=\sum_{d}mi\left(w,d\right)
\]
\end_inset
and the fractional right marginal mutual information is
\begin_inset Formula
\[
MI_{\mbox{right}}\left(w\right)=\frac{mi_{\mbox{right}}\left(w\right)}{P\left(w,*\right)}
\]
\end_inset
The grand-total MI for the dataset is
\begin_inset Formula
\[
MI=MI_{\mbox{tot}}=\sum_{w,d}mi\left(w,d\right)=\sum_{w}P\left(w,*\right)MI_{\mbox{right}}\left(w\right)=\sum_{d}P\left(*,d\right)MI_{\mbox{left}}\left(d\right)
\]
\end_inset
As noted earlier, the mutual information is minus the total entropy, up
to entropies of the marginal probabilities:
\begin_inset Formula
\[
MI_{\mbox{tot}}=S_{\mbox{right}}+S_{\mbox{left}}-H_{\mbox{tot}}
\]
\end_inset
\end_layout
\begin_layout Subsection*
Dataset Summary
\end_layout
\begin_layout Standard
Starting point: the dataset `r9-sim-200.rdb` after running `compute-mi` on
it, to get the full collection of stats on (w,d) pairs.
The stat summary is:
\end_layout
\begin_layout Standard
\begin_inset VSpace defskip
\end_inset
\end_layout
\begin_layout Standard
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="2" columns="5">
<features tabularvalignment="middle">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
dataset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $H_{\mbox{tot}}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $S_{\mbox{left}}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $S_{\mbox{right}}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $MI_{\mbox{tot}}$
\end_inset
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
r9-sim-200+mi.rdb
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
19.463
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
16.535
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
8.5148
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
5.5865
\end_layout
\end_inset
</cell>
</row>
</lyxtabular>
\end_inset
\end_layout
\begin_layout Standard
\begin_inset VSpace defskip
\end_inset
\end_layout
\begin_layout Standard
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="2" columns="5">
<features tabularvalignment="middle">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
dataset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $N_{\mbox{words}}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $N_{\mbox{dj}}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $N_{\mbox{pairs}}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $N\left(*,*\right)$
\end_inset
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
r9-sim-200+mi.rdb
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
15083
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
1043583
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
2777968
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
22942644
\end_layout
\end_inset
</cell>
</row>
</lyxtabular>
\end_inset
\begin_inset VSpace defskip
\end_inset
\end_layout
\begin_layout Standard
Here,
\begin_inset Formula $N_{\mbox{pairs}}$
\end_inset
is the total number of pairs
\begin_inset Formula $\left(w,d\right)$
\end_inset
that were observed (i.e.
pairs with non-zero counts).
The total number of observations of these pairs is
\begin_inset Formula $N\left(*,*\right)$
\end_inset
.
The sparsity is
\begin_inset Formula $-\log_{2}N_{\mbox{pairs}}/\left(N_{\mbox{words}}\times N_{\mbox{dj}}\right)=12.468$
\end_inset
.
\end_layout
\begin_layout Subsection*
Zipf Distribution
\end_layout
\begin_layout Standard
Some graphs follow.
\begin_inset Foot
status collapsed
\begin_layout Plain Layout
The data was processed with the `utils/entropy-marginals.scm` scripts in
this directory.
\end_layout
\end_inset
We begin with a very conventional graph, and a very conventional result.
This shows the marginal frequency
\begin_inset Formula $P\left(w,*\right)$
\end_inset
as a function of the rank.
The distribution is almost Zipfian, with a classical Zipf slope.
It's a bit noisier than hoped for; the noise may be due to the somewhat
imperfect dataset (it contains a lot of words with escaped backslashes
in them, due to a very early parsing bug.)
\end_layout
\begin_layout Standard
\align center
\begin_inset Graphics
filename p5-entropy/rank-wfreq.eps
width 80text%
\end_inset
\end_layout
\begin_layout Standard
A similar figure can be drawn for the entropy; it is very nearly identical.
\end_layout
\begin_layout Subsection*
Entropy Distributions
\end_layout
\begin_layout Standard
Next: a histogram of
\begin_inset Formula $H_{\mbox{right}}\left(w\right)$
\end_inset
for the 15083 words
\begin_inset Formula $w$
\end_inset
in the dataset.
These are placed into 200 bins and counted with a weight of 1.0 per word.
\end_layout
\begin_layout Standard
\align center
\begin_inset Graphics
filename p5-entropy/bin-went.eps
width 80text%
\end_inset
\end_layout
\begin_layout Standard
The smooth curve marked G is the Gaussian distribution
\begin_inset Formula
\[
G\left(x;\mu,\sigma\right)=\frac{1}{\sigma\sqrt{2\pi}}\exp-\frac{\left(x-\mu\right)^{2}}{2\sigma^{2}}
\]
\end_inset
This fit is not particularly convincing; there's a sharp drop-off to the
right that is not being fit.
The data is fairly noisy.
My guess is that the spiky bits on the right are from infrequently-observed
word-disjunct pairs.
\end_layout
\begin_layout Standard
This suggests redoing the bin-counts, weighting each word by it's probability.
This is shown below.
\end_layout
\begin_layout Standard
\align center
\begin_inset Graphics
filename p5-entropy/bin-wei-went.eps
width 80col%
\end_inset
\end_layout
\begin_layout Standard
This shows the same data, with each word weighted by the probability of
that word.
That is, this is a graph of
\begin_inset Formula $H_{\mbox{right}}\left(w\right)$
\end_inset
with each point weighted by
\begin_inset Formula $P\left(w,*\right)$
\end_inset
.
This makes it clear that the right side of the picture, the high entropy
side, is associated with infrequent words.
\end_layout
\begin_layout Standard
Also, this time, the Gaussian is an excellent fit.
Note that the Gaussian is centered on
\begin_inset Formula $H_{\mbox{tot}}=19.463$
\end_inset
for the dataset.
This is as it should be: the total entropy is the average of this distribution.
\end_layout
\begin_layout Subsection*
MI Distributions
\end_layout
\begin_layout Standard
Same as above, but for the word-disjunct mutual information.
\end_layout
\begin_layout Standard
\align center
\begin_inset Graphics
filename p5-entropy/bin-wei-wmi.eps
width 80text%
\end_inset
\end_layout
\begin_layout Standard
It seems reasonable to expect a Gaussian distribution here, also, but it
appears to not be the case.
A log-linear fit seems better.
It's not clear whether this should be dismissed as
\begin_inset Quotes eld
\end_inset
bad data
\begin_inset Quotes erd
\end_inset
or not.
Note that the Gaussian is centered on
\begin_inset Formula $MI_{\mbox{tot}}=5.5865$
\end_inset
.
The histogram has been manually checked, and integrates to this same value.
That is might be the case is not obvious in this semi-log plot, but the
very high frequency of values
\begin_inset Formula $MI<5$
\end_inset
is enough to balance the long tail on the right.
\end_layout
\begin_layout Subsection*
Scatter-plots
\end_layout
\begin_layout Standard
How does the MI correlate with the entropy and the frequency? Three scatter-plot
s reveal this relationship.
The first scatter-plot shows the entropy
\begin_inset Formula $H_{\mbox{right}}\left(w\right)$
\end_inset
vs.
the log of the marginal frequency
\begin_inset Formula $-\log_{2}P\left(w,*\right)$
\end_inset
.
The two straight lines have the slopes as indicated, and approximately
bound the distribution.
\end_layout
\begin_layout Standard
\align center
\begin_inset Graphics
filename p5-entropy/scat-wfreq-went.eps
width 80col%
\end_inset
\end_layout
\begin_layout Standard
The graph below shows the mutual information
\begin_inset Formula $MI_{\mbox{right}}\left(w\right)$
\end_inset
vs.
the log of the marginal frequency
\begin_inset Formula $-\log_{2}P\left(w,*\right)$
\end_inset
.
Again, two lines approximately bound the distribution.
Note that the average MI of this distribution is
\begin_inset Formula $MI_{\mbox{tot}}=5.5865$
\end_inset
.
Thus, we have a very long tail offsetting a very heavy low-MI, high-frequency
stub.
Similarly, the average of
\begin_inset Formula $-\log_{2}P\left(w,*\right)$
\end_inset
is
\begin_inset Formula $S_{\mbox{right}}=8.5148$
\end_inset
.
Thus, the centroid of this scatter-plot is in the lower-left, and not where
one might visually expect it.
\end_layout
\begin_layout Standard
\align center
\begin_inset Graphics
filename p5-entropy/scat-wfreq-wmi.eps
width 80col%
\end_inset
\end_layout
\begin_layout Standard
Finally, the the entropy
\begin_inset Formula $H_{\mbox{right}}\left(w\right)$
\end_inset
vs.
he mutual information
\begin_inset Formula $MI_{\mbox{right}}\left(w\right)$
\end_inset
.
Rather than attempting to bound this distribution, the two lines attempt
to bisect it.
This is an
\begin_inset Quotes eld
\end_inset
unweighted
\begin_inset Quotes erd
\end_inset
bisection.
The actual centroid of this figure is at
\begin_inset Formula $H_{\mbox{tot}}=19.463$
\end_inset
and
\begin_inset Formula $MI_{\mbox{tot}}=5.5865$
\end_inset
, at the lower middle of this scatter-plot.
\end_layout
\begin_layout Standard
\align center
\begin_inset Graphics
filename p5-entropy/scat-went-wmi.eps
width 80col%
\end_inset
\end_layout
\begin_layout Standard
In conclusion: there is an obvious correlation between frequency and entropy,
and frequency and MI.
However, the correlation between entropy and MI is considerably weaker.
\end_layout
\begin_layout Subsection*
Alternative Rankings
\end_layout
\begin_layout Standard
The Diary Part Three proposes that two words should form the seed of a cluster,
if they have the largest ranked-MI of all word-pairs.
Can the above provide some alternative rankings? First, recall the definition
of the ranked-MI.
\end_layout
\begin_layout Subsubsection*
Ranked-MI
\end_layout
\begin_layout Standard
The following is a quick recap of the ranked-MI, as discovered and developed
in Parts Three and Four of the Diary.
The ranked-MI was built out of the common-MI plus Q, as follows.
Specifically, this repeats and condenses the content of Part Three, pages
18-19, as well as Diary Part Four, page 19.
Start by defining
\begin_inset Formula $N\left(w,d\right)$
\end_inset
as the observation count on the word-disjunct pair
\begin_inset Formula $\left(w,d\right)$
\end_inset
.
From this, the observation frequency follows as
\begin_inset Formula
\[
P\left(w,d\right)=\frac{N\left(w,d\right)}{N\left(*,*\right)}
\]
\end_inset
The joint frequency of two words
\begin_inset Formula $w,u$
\end_inset
is then defined as the matrix product
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
f\left(w,u\right)=\sum_{d}P\left(w,d\right)P\left(u,d\right)
\]
\end_inset
This is not normalized, in that
\begin_inset Formula $f\left(*,*\right)\ne1$
\end_inset
which leads to the definition of
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
Q=-\log_{2}f\left(*,*\right)=-\log_{2}\sum_{d}P\left(*,d\right)P\left(*,d\right)
\]
\end_inset
which has a value of
\begin_inset Formula $Q=11.945777$
\end_inset
for this dataset.
Note the minus sign.
This means
\begin_inset Formula $1\gg f\left(*,*\right)$
\end_inset
.
The marginals can then be derived as
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
f\left(w\right)=f\left(w,*\right)=\sum_{d}P\left(w,d\right)P\left(*,d\right)
\]
\end_inset
The word-pair MI is defined as
\begin_inset Formula
\[
MI\left(u,w\right)=\log_{2}\frac{f\left(w,u\right)f\left(*,*\right)}{f\left(w\right)f\left(u\right)}
\]
\end_inset
Note the definition above is
\begin_inset Quotes eld
\end_inset
projective
\begin_inset Quotes erd
\end_inset
, in that it is independent of the overall normalization of
\begin_inset Formula $f$
\end_inset
.
This is seen explicitly, by expanding out:
\begin_inset Formula
\begin{align*}
MI\left(u,w\right)= & \log_{2}\frac{\left[\sum_{d}P\left(u,d\right)P\left(w,d\right)\right]\left[\sum_{d}P\left(*,d\right)P\left(*,d\right)\right]}{\left[\sum_{d}P\left(u,d\right)P\left(*,d\right)\right]\left[\sum_{d}P\left(w,d\right)P\left(*,d\right)\right]}\\
= & \log_{2}\frac{\left[\sum_{d}N\left(u,d\right)N\left(w,d\right)\right]\left[\sum_{d}N\left(*,d\right)N\left(*,d\right)\right]}{\left[\sum_{d}N\left(u,d\right)N\left(*,d\right)\right]\left[\sum_{d}N\left(w,d\right)N\left(*,d\right)\right]}
\end{align*}
\end_inset
The ranked-MI is defined as
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{align*}
\mbox{ranked}MI\left(w,u\right)= & MI\left(u,w\right)+\frac{1}{2}\log_{2}f\left(u\right)+\frac{1}{2}\log_{2}f\left(w\right)+Q\\
= & \log_{2}\frac{f\left(w,u\right)}{\sqrt{f\left(w\right)f\left(u\right)}}
\end{align*}
\end_inset
Just like the MI, the ranked-MI is scale invariant; its projective.
This suggests that it has some interesting theory behind it.
Graphs for the ranked-MI for this dataset can be found at the end of Part
Three.
They are pretty Gaussians.
\end_layout
\begin_layout Subsection*
Probabilities over Word Pairs
\end_layout
\begin_layout Standard
The ratio
\begin_inset Formula
\[
\pi\left(u,w\right)=\frac{f\left(w,u\right)}{f\left(*,*\right)}
\]
\end_inset
is a probability.
Thus, we can play the game again, this time with
\begin_inset Formula $\pi$
\end_inset
and ask about conditional probabilities, mutual information and entropies.
For example, what is the value of
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
RMI_{\mbox{tot}}=\sum_{w,u}\pi\left(w,u\right)\log_{2}\frac{\pi\left(w,u\right)}{\sqrt{\pi\left(w\right)\pi\left(u\right)}}
\]
\end_inset
This and related questions have to stay out of reach, however, since computing
\begin_inset Formula $\pi\left(u,w\right)$
\end_inset
for all word-pairs
\begin_inset Formula $\left(u,w\right)$
\end_inset
is computationally infeasible at this time.
(Well, it could be done
\begin_inset Quotes eld
\end_inset
easily
\begin_inset Quotes erd
\end_inset
on a cloud instance with a few hundred CPU's, but I don't have that, or
even the code to leverage that.)
\end_layout
\begin_layout Subsubsection*
Alternative Rankings
\end_layout
\begin_layout Standard
Following the above, several variations suggest themselves:
\begin_inset Formula
\[
HMI\left(u,w\right)=MI\left(u,w\right)+\frac{1}{2}H_{\mbox{right}}\left(u\right)+\frac{1}{2}H_{\mbox{right}}\left(w\right)
\]
\end_inset
and
\begin_inset Formula
\[
MMI\left(u,w\right)=MI\left(u,w\right)+\frac{1}{2}MI_{\mbox{right}}\left(u\right)+\frac{1}{2}MI_{\mbox{right}}\left(w\right)
\]
\end_inset
Either of these feel like promising improvements to ranked-MI, as, experimentall
y, ranked-MI is proving to be less than reliable for merge determinations.
As mentioned earlier, the core problem is that ranked-MI seems to be suggesting
that clusters should be merged far too often, presumably because
\begin_inset Formula $\frac{1}{2}\log_{2}f\left(w\right)$
\end_inset
gets larger and larger as the clusters grow.
\end_layout
\begin_layout Standard
Neither of the above have
\begin_inset Quotes eld
\end_inset
pretty
\begin_inset Quotes erd
\end_inset
interpretations, which can be seen by recalling the definitions above:
\begin_inset Formula
\[
H_{\mbox{right}}\left(w\right)=-\frac{\sum_{d}P\left(w,d\right)\log_{2}P\left(w,d\right)}{P\left(w,*\right)}
\]
\end_inset
\end_layout
\begin_layout Standard
and
\begin_inset Formula
\[
MI_{\mbox{right}}\left(w\right)=\frac{1}{P\left(w,*\right)}\sum_{d}P\left(w,d\right)\log_{2}\frac{P\left(w,d\right)}{P\left(*,d\right)P\left(w,*\right)}
\]
\end_inset
These are less compelling because of this complexity.
\end_layout
\begin_layout Subsection*
Alternative Similarity Distributions
\end_layout
\begin_layout Standard
We previously saw that common-MI had a nice Gaussian distribution.
Is this still the case for HMI and MMI? Lets find out ...
\end_layout
\begin_layout Standard
In Part Three, the graphs were done with a set of similarities for the top
1200 words, computed without shapes.
What I have handy for the current dataset is much smaller: just the top
200 most frequent words; however, this time, similarities were computed
using shapes.
So its ..
different.
Anyway, there's a total of
\begin_inset Formula $N\left(N+1\right)/2=20100$
\end_inset
similarities handy.
Of these, 175 do not have an MI; that is, the dot-product is zero, so the
MI is
\begin_inset Formula $-\infty$
\end_inset
.
Thus, only 20100-175=19925 actual pairs.
\end_layout
\begin_layout Standard
Four nearly identical graphs, differing only in the horizontal offset:
\end_layout
\begin_layout Standard
\align center
\begin_inset Graphics
filename p5-sim/sim-mi.eps
width 50text%
\end_inset
\begin_inset Graphics
filename p5-sim/sim-rmi.eps
width 50text%
\end_inset
\end_layout
\begin_layout Standard
\align center
\begin_inset Graphics
filename p5-sim/sim-hmi.eps
width 50text%
\end_inset
\begin_inset Graphics
filename p5-sim/sim-mmi.eps
width 50text%
\end_inset
\end_layout
\begin_layout Standard
The Gaussians are an eyeballed fit.
Perhaps there are some popular fat-tailed distributions that would give
a better fit? At any rate, the fat-tail is on the left, an excessive of
word-pairs judged to be dis-similar.
Perhaps this is a good thing? That is, the MPG parsing gave us a bunch
of disjuncts that are
\begin_inset Quotes eld
\end_inset
obviously
\begin_inset Quotes erd
\end_inset
dis-similar, which is exactly what we hope to see.
Perhaps the fatness of the tail is correlated with the number of distinct
word-senses? That is, we expect words with different word-senses to be
dis-similar to one-another, and so the more word-senses there are, the
fatter the tail? What is the theoretical exposition for this effect?
\end_layout
\begin_layout Standard
How similar are these graphs? All four are re-plotted below, with the horizontal
offset adjusted by the eyeballed fits, above.
\end_layout
\begin_layout Standard
\align center
\begin_inset Graphics
filename p5-sim/simil-all.eps
width 80text%
\end_inset
\end_layout
\begin_layout Standard
Clearly, these are almost identical, but not quite.
Given the earlier scatter-plots, the specific rankings of specific pairs
must be different in each of these.
How different? The table below lists the top 17 most similar pairs for
each style of similarity.
The top number is the similarity score for each style; the negative numbers
are how much less the next score is, from the top score.
\end_layout
\begin_layout Standard
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\end_inset
\end_layout
\begin_layout Standard
\align center
\begin_inset Tabular
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\end_inset
\end_layout
\begin_layout Standard
\begin_inset VSpace defskip
\end_inset
What can we conclude? Clearly, the naked MI score is exploring all pairings
of similar things.
By using the in-group algo, we could expect that all the punctuation gets
lumped together.
The HMI seems to fall into a similar trap: the in-group seems to be
\begin_inset Quotes eld
\end_inset
their our its your his my
\begin_inset Quotes erd
\end_inset
.
\end_layout
\begin_layout Standard
Hard to figure out what to make of this.
\end_layout
\begin_layout Standard
A tempting conclusion, perhaps a bit
\emph on
ad hoc
\emph default
, is to use ranked-MI to suggest the initial pair to seed a cluster, but
then to use either MI, HMI or MMI to expand the actual in-group.
So, MMI looks pretty slick, but is computationally the heaviest to recompute.
HMI should be fairly fast, as it's just some logs.
And MI is computationally
\begin_inset Quotes eld
\end_inset
free
\begin_inset Quotes erd
\end_inset
, it has to be computed anyway.
\end_layout
\begin_layout Standard
Where to change this: search code for `optimal-in-group` and specify the
desired similarity function.
\end_layout
\begin_layout Standard
Hypothesis: it is also possible that these different in-group suggestions
won't alter very much.
This is because it is not the initial pair that matters, it is the in-group
near that initial pair that matters.
Perhaps all of these end up suggesting similar, or the same in-groups?
\end_layout
\begin_layout Section*
Experiment-11 – Deep Merge (also Expt 12 graphs)
\end_layout
\begin_layout Standard
Given the above results, it seems time for a deeper-merge experiment.
Unlike experiment-10, this one focuses on:
\end_layout
\begin_layout Itemize
Using plain-MI instead of rank-MI for determining the in-group.
\end_layout
\begin_layout Itemize
Fixes quorum=0.7 and commonality=0.2 and varies the noise: noise= 4,3,2,1
\end_layout
\begin_layout Standard
Based on expt-10 it looks like quorum=0.7 or 0.8 would give better results.
Also, it seems like commonality has almost no effect: the chosen grouping
pretty much always is arrived at with the back-off strategy, rather than
with commonality exceeding the threshold.
By contrast, the noise-threshold appears to have a huge effect on what
gets merged.
Here's an example.
For the very first merge, we see
\begin_inset VSpace defskip
\end_inset
\end_layout
\begin_layout Standard
\begin_inset listings
inline false
status open
\begin_layout Plain Layout
------ Round 1 Next in line:
\end_layout
\begin_layout Plain Layout
ranked-MI = 9.1671 MI = 9.3888 (`—`, `+`)
\end_layout
\begin_layout Plain Layout
------ Start merge 1 with seed pair `—` and `+` Initial in-group size=8:
`+` `—` `”` `_` `)` `[` `(` `]`
\end_layout
\begin_layout Plain Layout
In-group size=8 overlap = 9 of 10975 disjuncts, commonality= 0.08%
\end_layout
\begin_layout Plain Layout
In-group size=7 overlap = 17 of 10530 disjuncts, commonality= 0.16%
\end_layout
\begin_layout Plain Layout
In-group size=6 overlap = 33 of 10107 disjuncts, commonality= 0.33%
\end_layout
\begin_layout Plain Layout
In-group size=5 overlap = 29 of 9433 disjuncts, commonality= 0.31%
\end_layout
\begin_layout Plain Layout
In-group size=6: `+` `—` `”` `_` `)` `[`
\end_layout
\end_inset
\begin_inset VSpace defskip
\end_inset
Notable is just how pathetically low the commonality is.
However, when the merge is actually performed, we see something quite different.
For noise=4, we get
\begin_inset VSpace defskip
\end_inset
\end_layout
\begin_layout Standard
\begin_inset listings
inline false
status open
\begin_layout Plain Layout
------ merge-majority: Merge 5550 of 7957 sections in 12 secs
\end_layout
\begin_layout Plain Layout
------ merge-majority: Remaining 14284 of 22825 cross in 43 secs
\end_layout
\end_inset
\begin_inset VSpace defskip
\end_inset
which tells us that almost all of the merged sections had an observation
count of less than or equal to 4! That is, 5550+14284 were merged, whereas
we expected only 6 x 33 x 70% = 140 to get merged! Yowie! This means that
only 140 sections/cross-sections had an observation count of greater than
4; the rest of them had a lesser count, and were merged only because they
were swept up as being under the noise floor.
\end_layout
\begin_layout Standard
For noise=3, we see that the number of merged sections drops:
\begin_inset VSpace defskip
\end_inset
\begin_inset listings
inline false
status open
\begin_layout Plain Layout
------ merge-majority: Merge 4884 of 7957 sections in 11 secs
\end_layout
\begin_layout Plain Layout
------ merge-majority: Remaining 12502 of 22863 cross in 37 secs
\end_layout
\end_inset
\begin_inset VSpace defskip
\end_inset
\end_layout
\begin_layout Standard
For noise=2, the number of merged sections drops again.
It's tempting to say that it's
\begin_inset Quotes eld
\end_inset
surprisingly high
\begin_inset Quotes erd
\end_inset
, but its not: The Zipfian distribution tells us that there's an incredibly
fat tail at these low counts.
\begin_inset VSpace defskip
\end_inset
\begin_inset listings
inline false
status open
\begin_layout Plain Layout
------ merge-majority: Merge 3586 of 7957 sections in 10 secs
\end_layout
\begin_layout Plain Layout
------ merge-majority: Remaining 9135 of 22963 cross in 35 secs
\end_layout
\end_inset
\begin_inset VSpace defskip
\end_inset
\end_layout
\begin_layout Standard
How much harm is there in merging of these low-count tails? My current guess
is
\begin_inset Quotes eld
\end_inset
none
\begin_inset Quotes erd
\end_inset
.
These low-count tails contribute very little, or nothing to the word-pair
MI.
Right? Or are they nickel-n-diming us to death? (Hmm.
If they contribute nothing, then they could have been trimmed away from
the get-go...) Mostly, they hang around, chewing up space...
Later on, when a dictionary is created, these can be trimmed away, to arrive
at a more compact dictionary.
I don't have insight here.
It's confusing.
\end_layout
\begin_layout Standard
On the other hand, due to the preferential-attachment phenomenon that is
being seen (large clusters getting larger) it might be that these fat tails
are wagging the dog: they suck in the noise, and then this noise sucks
in other tails.
These tails might end up dominating the sense of the cluster? Perhaps we
should have a noise=0 run to compare against?
\end_layout
\begin_layout Standard
At any rate, its kind of surprising that the MI can be high but the commonality
is so low.
This doesn't make sense intuitively.
\end_layout
\begin_layout Subsection*
What contributes to the MI?
\end_layout
\begin_layout Standard
The last sentence immediately above seems important enough to get it's own
subsection.
What, exactly, is contributing to the MI? We see high MI values, but a
low Jaccard similarity.
This seems surprising.
Is it because the MI's are pairwise, while the Jaccard is for the group
as a whole?
\end_layout
\begin_layout Standard
How does the low-count tail influence the MI? When does trimming tails help?
When does it hurt?
\end_layout
\begin_layout Subsection*
Progress graphs
\end_layout
\begin_layout Standard
Far more graphs from this run.
All of these graphs are for datasets for which quorum=0.7 and commonality=0.2.
They differ in the noise setting, varying from 4 to 1.
The
\begin_inset Quotes eld
\end_inset
older
\begin_inset Quotes erd
\end_inset
dataset is from the previous run, way up above, which had quorum=0.7, commonalit
y=0.4 and noise=4.
Note that assorted buglets were fixed since that older run.
\end_layout
\begin_layout Standard
The
\begin_inset Quotes eld
\end_inset
precise
\begin_inset Quotes erd
\end_inset
dataset recomputes all word-pair MI for every possible word that was affected
by the merge.
This can be hundreds of words, thousands of word-pairs (and is thus slow
to compute).
The
\begin_inset Quotes eld
\end_inset
imprecise
\begin_inset Quotes erd
\end_inset
datasets recompute the MI only for the words directly involved in the merge
(so, for only a handful of words, and so much faster to compute.) Thus,
one of the questions is whether the
\begin_inset Quotes eld
\end_inset
precise
\begin_inset Quotes erd
\end_inset
datasets provide materially better classifications, or not.
\end_layout
\begin_layout Standard
Computation of these datasets was terminated with a snowstorm power outage.
Rather than resuming where these left off, a fresh computation was started,
with a slightly cleaned up dataset, Experiment 12.
As can be seen, there are no substantial changes arising from this.
\end_layout
\begin_layout Standard
Shown for comparison are the Experiment-12 graphs.
These are based on the same underlying dataset as Experiment-11, but are
cleaned up to remove unconnectable words and disjuncts.
That is, the dataset is scrubbed to remove all connectors that cannot connect
to words, and vice-versa.
This has a minor effect on the graphs, but reduces the amount of annoyance
and confusion during inspection and debugging.
\end_layout
\begin_layout Standard
The Experiment-12
\begin_inset Quotes eld
\end_inset
imprecise
\begin_inset Quotes erd
\end_inset
runs all failed when they ran out of hugepage memory.
Foo.
Unable to recover and pick up where they left off; there is some kind of
bug that prevents restarting (the marginals get unbalanced, and recomputing
them leaves something wrong...) These halted as shown:
\begin_inset VSpace defskip
\end_inset
\end_layout
\begin_layout Standard
\align center
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1343
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\family sans
r12-log-q0.7-c0.2-n2.dat
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
891
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\family sans
r12-log-q0.7-c0.2-n3.dat
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
1062
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\family sans
r12-log-q0.7-c0.2-n4.dat
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
1058
\end_layout
\end_inset
</cell>
</row>
</lyxtabular>
\end_inset
\begin_inset VSpace defskip
\end_inset
\end_layout
\begin_layout Standard
The last two stopped at almost the same place, and thus allow head-to-head
comparisons.
See much farther below.
\end_layout
\begin_layout Subsubsection*
Sparsity
\end_layout
\begin_layout Standard
Sparsity is defined as always: it is log2 of the fraction of non-zero matrix
entries to total possible matrix entries.
Gut instinct says 'slower sparsity growth is better'.
From the first graph, though, we find a very non-linear growth rate depending
on the noise threshold.
\end_layout
\begin_layout Standard
\align center
\begin_inset Graphics
filename p5-run-11-merge/r11-sparsity.eps
width 50text%
\end_inset
\begin_inset Graphics
filename p5-run-11-merge/r11-p-sparsity.eps
width 50text%
\end_inset
\end_layout
\begin_layout Standard
\align center
\begin_inset Graphics
filename p5-run-12-merge/r12-sparsity.eps
width 50text%
\end_inset
\begin_inset Graphics
filename p5-run-12-merge/r12-p-sparsity.eps
width 50text%
\end_inset
\end_layout
\begin_layout Standard
For the new improved cleaned-up dataset, the initial sparsity is lower.
\end_layout
\begin_layout Subsubsection*
MMT Entropy
\end_layout
\begin_layout Standard
The
\begin_inset Formula $MM^{T}$
\end_inset
-entropy is as defined before; it is
\begin_inset Formula
\[
H=\log_{2}
\]
\end_inset
\end_layout
\begin_layout Standard
where
\begin_inset Formula
\[
f\left(w,u\right)=\sum_{d}P\left(w,d\right)P\left(u,d\right)
\]
\end_inset
\end_layout
\begin_layout Standard
\align center
\begin_inset Graphics
filename p5-run-11-merge/r11-entropy.eps
width 50text%
\end_inset
\begin_inset Graphics
filename p5-run-11-merge/r11-p-entropy.eps
width 50text%
\end_inset
\end_layout
\begin_layout Standard
\align center
\begin_inset Graphics
filename p5-run-12-merge/r12-entropy.eps
width 50text%
\end_inset
\begin_inset Graphics
filename p5-run-12-merge/r12-p-entropy.eps
width 50text%
\end_inset
\end_layout
\begin_layout Standard
Note that the cleanup has almost no impact on the MMT Entropy.
It is slightly different, but not by much.
\end_layout
\begin_layout Subsubsection*
MMT-q
\end_layout
\begin_layout Standard
The
\begin_inset Formula $MM^{T}q$
\end_inset
is the
\begin_inset Formula $Q$
\end_inset
value as defined before, above, and in 'Diary Part Four':
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
Q=-\log_{2}\sum_{d}P\left(*,d\right)P\left(*,d\right)
\]
\end_inset
It varies over time, as both new disjuncts are created, old disjuncts have
counts drop to zero, and similarly, come words have counts drop to zero
as they are transferred to the word-classes.
\end_layout
\begin_layout Standard
\align center
\begin_inset Graphics
filename p5-run-11-merge/r11-mmtq.eps
width 50text%
\end_inset
\begin_inset Graphics
filename p5-run-11-merge/r11-p-mmtq.eps
width 50text%
\end_inset
\end_layout
\begin_layout Standard
\align center
\begin_inset Graphics
filename p5-run-12-merge/r12-mmtq.eps
width 50text%
\end_inset
\begin_inset Graphics
filename p5-run-12-merge/r12-p-mmtq.eps
width 50text%
\end_inset
\end_layout
\begin_layout Subsubsection*
Ranked-MI
\end_layout
\begin_layout Standard
The next graph shows the ranked-MI of the top-ranked word-pair that triggers
the merge.
Were it not for the re-computations of word-pair MI's, this should trend
downwards, monotonically.
The re-computation of MI exposes new fragments/remainders of merged classes
that elevate this MI, briefly, until undisturbed word-pairs are encountered
again.
\end_layout
\begin_layout Standard
Ranked-MI is defined as in 'Diary Part Four', and restated earlier, above.
\end_layout
\begin_layout Standard
\align center
\begin_inset Graphics
filename p5-run-11-merge/r11-ranked-mi.eps
width 50text%
\end_inset
\begin_inset Graphics
filename p5-run-11-merge/r11-p-ranked-mi.eps
width 50text%
\end_inset
\end_layout
\begin_layout Standard
\align center
\begin_inset Graphics
filename p5-run-12-merge/r12-ranked-mi.eps
width 50text%
\end_inset
\begin_inset Graphics
filename p5-run-12-merge/r12-p-ranked-mi.eps
width 50text%
\end_inset
\end_layout
\begin_layout Subsubsection*
MI
\end_layout
\begin_layout Standard
The ranked-MI is build as a linear combination of the word-pair MI, plus
the log of the frequencies, plus Q.
It seems reasonable to ask what the MI was of the pair that initiated the
merge.
This is shown below.
\end_layout
\begin_layout Standard
\align center
\begin_inset Graphics
filename p5-run-11-merge/r11-plain-mi.eps
width 80text%
\end_inset
\end_layout
\begin_layout Subsubsection*
Marginal Entropies
\end_layout
\begin_layout Standard
The next four graphs require the frequencies
\begin_inset Formula $P\left(w,d\right)$
\end_inset
to be computed.
This turns out to be error-prone and challenging, and thus will be reported
here, but turned off in future versions of the code (can be re-enabled
by setting
\family sans
(define TRACK-ENTROPY #t)
\family default
in the code.).
The problems are these:
\end_layout
\begin_layout Itemize
The pair-frequencies must be recomputed before starting, and stored.
This includes the frequencies on CrossSections, which are not normally
stored.
After this, the marginal frequencies and the marginal entropies must be
computed.
\end_layout
\begin_layout Itemize
As the merges proceed, entropies must be recomputed (they are, if the flag
is set).
This adds approx 5 minutes to each merge cycle, which is a significant
overhead.
\end_layout
\begin_layout Itemize
The alternative is to create a new matrix API for entropies that always
computes what is needed on-the-fly, from the counts (which are always correct).
Doing this is ...
OK, as it will reduce operational errors, but will add yet more CPU overhead
to obtaining these stats.
\end_layout
\begin_layout Standard
Given these issues, it seems that tracking entropies is not worth it.
I mean, the graphs below are interesting, and all, but they are not Earth-shaki
ng, and don't seem to offer any new insight, beyond some warm fuzzies that
everything is going fine.
\end_layout
\begin_layout Standard
These remarks apply to some of the class entropies, much further below.
They won't be available because of the dependency on the
\family sans
pair-freq-api
\family default
object.
\end_layout
\begin_layout Subsubsection*
Marginal Word Entropy
\end_layout
\begin_layout Standard
The right-marginal or word entropy is as defined above:
\begin_inset Formula
\[
S_{\mbox{right}}=\sum_{w}P\left(w,*\right)\log_{2}P\left(w,*\right)
\]
\end_inset
Yes, this probably should have been called
\begin_inset Quotes eld
\end_inset
left
\begin_inset Quotes erd
\end_inset
, but that introduces new confusions.
\end_layout
\begin_layout Standard
\align center
\begin_inset Graphics
filename p5-run-11-merge/r11-right-dj-entropy.eps
width 50text%
\end_inset
\begin_inset Graphics
filename p5-run-11-merge/r11-p-right-dj-entropy.eps
width 50text%
\end_inset
\end_layout
\begin_layout Standard
\align center
\begin_inset Graphics
filename p5-run-12-merge/r12-right-dj-entropy.eps
width 50text%
\end_inset
\begin_inset Graphics
filename p5-run-12-merge/r12-p-right-dj-entropy.eps
width 50text%
\end_inset
\end_layout
\begin_layout Subsubsection*
Marginal disjunct entropy
\end_layout
\begin_layout Standard
As defined above:
\begin_inset Formula
\[
S_{\mbox{left}}=\sum_{d}P\left(*,d\right)\log_{2}P\left(*,d\right)
\]
\end_inset
\end_layout
\begin_layout Standard
\align center
\begin_inset Graphics
filename p5-run-11-merge/r11-left-dj-entropy.eps
width 50text%
\end_inset
\begin_inset Graphics
filename p5-run-11-merge/r11-p-left-dj-entropy.eps
width 50text%
\end_inset
\end_layout
\begin_layout Standard
\align center
\begin_inset Graphics
filename p5-run-12-merge/r12-left-dj-entropy.eps
width 50text%
\end_inset
\begin_inset Graphics
filename p5-run-12-merge/r12-p-left-dj-entropy.eps
width 50text%
\end_inset
\end_layout
\begin_layout Subsubsection*
Total Entropy
\end_layout
\begin_layout Standard
The total entropy is as defined above
\begin_inset Formula
\[
H_{\mbox{tot}}=-\sum_{w,d}P\left(w,d\right)\log_{2}P\left(w,d\right)
\]
\end_inset
\end_layout
\begin_layout Standard
\align center
\begin_inset Graphics
filename p5-run-11-merge/r11-word-dj-entropy.eps
width 50text%
\end_inset
\begin_inset Graphics
filename p5-run-11-merge/r11-p-word-dj-entropy.eps
width 50text%
\end_inset
\end_layout
\begin_layout Standard
\align center
\begin_inset Graphics
filename p5-run-12-merge/r12-word-dj-entropy.eps
width 50text%
\end_inset
\begin_inset Graphics
filename p5-run-12-merge/r12-p-word-dj-entropy.eps
width 50text%
\end_inset
\end_layout
\begin_layout Subsubsection*
Total MI
\end_layout
\begin_layout Standard
The total MI is as defined above:
\begin_inset Formula
\begin{align*}
MI_{\mbox{tot}}= & \sum_{w,d}P\left(w,d\right)\log_{2}\frac{P\left(w,d\right)}{P\left(*,d\right)P\left(w,*\right)}\\
= & S_{\mbox{right}}+S_{\mbox{left}}-H_{\mbox{tot}}
\end{align*}
\end_inset
\end_layout
\begin_layout Standard
\align center
\begin_inset Graphics
filename p5-run-11-merge/r11-word-dj-mi.eps
width 50text%
\end_inset
\begin_inset Graphics
filename p5-run-11-merge/r11-p-word-dj-mi.eps
width 50text%
\end_inset
\end_layout
\begin_layout Standard
\align center
\begin_inset Graphics
filename p5-run-12-merge/r12-word-dj-mi.eps
width 50text%
\end_inset
\begin_inset Graphics
filename p5-run-12-merge/r12-p-word-dj-mi.eps
width 50text%
\end_inset
\end_layout
\begin_layout Subsubsection*
Number of Words per Class
\end_layout
\begin_layout Standard
As mergers proceed, it is not uncommon to lump additional words into an
existing class (possibly splitting that class into two).
Thus, over time, one expects the size of classes to grow.
This is charted below.
It shows the size of the word-class created at that time-step.
\end_layout
\begin_layout Standard
\align center
\begin_inset Graphics
filename p5-run-11-merge/r11-cls-nwords.eps
width 50text%
\end_inset
\begin_inset Graphics
filename p5-run-12-merge/r12-cls-nwords.eps
width 50text%
\end_inset
\end_layout
\begin_layout Subsubsection*
Self-MI of a Class
\end_layout
\begin_layout Standard
The MI of two words is as defined above:
\begin_inset Formula
\[
MI\left(u,w\right)=\log_{2}\frac{f\left(w,u\right)f\left(*,*\right)}{f\left(w\right)f\left(u\right)}
\]
\end_inset
where
\begin_inset Formula
\[
f\left(w,u\right)=\sum_{d}P\left(w,d\right)P\left(u,d\right)
\]
\end_inset
The self-MI of a word
\begin_inset Formula $w$
\end_inset
is them
\begin_inset Formula $MI\left(w,w\right)$
\end_inset
.
The graph below shows the self-MI of the class created at that time-step.
\end_layout
\begin_layout Standard
\align center
\begin_inset Graphics
filename p5-run-11-merge/r11-cls-self-mi.eps
width 50text%
\end_inset
\begin_inset Graphics
filename p5-run-12-merge/r12-cls-self-mi.eps
width 50text%
\end_inset
\end_layout
\begin_layout Subsubsection*
Self-ranked-MI
\end_layout
\begin_layout Standard
As above, but this is the ranked-MI of the class against itself.
\end_layout
\begin_layout Standard
\align center
\begin_inset Graphics
filename p5-run-11-merge/r11-cls-self-rmi.eps
width 50text%
\end_inset
\begin_inset Graphics
filename p5-run-12-merge/r12-cls-self-rmi.eps
width 50text%
\end_inset
\end_layout
\begin_layout Subsubsection*
Support of the Class
\end_layout
\begin_layout Standard
The support of a word is the grand-total number of disjuncts assigned to
that word; likewise for a word-class.
In formulas, the support is
\begin_inset Formula
\[
\Delta\left(w\right)=\sum_{d}\delta\left(w,d\right)
\]
\end_inset
where
\begin_inset Formula
\[
\delta\left(w,d\right)=\begin{cases}
1 & \mbox{if }N\left(w,d\right)>0\\
0 & \mbox{otherwise}
\end{cases}
\]
\end_inset
Shown below is
\begin_inset Formula $\log_{2}\Delta\left(w\right)$
\end_inset
for
\begin_inset Formula $w$
\end_inset
the recently created class.
Thus, the larger this is, the greater the support.
\end_layout
\begin_layout Standard
It might be more edifying to subtract this from the total number of disjuncts:
this is mildly variable, approximately equal to
\begin_inset Formula $1.05\times10^{6}$
\end_inset
– that is, a little over a million disjuncts, or about
\begin_inset Formula $\log_{2}1.05\times10^{6}\approx20$
\end_inset
.
This would then give the fraction of all possible disjuncts appearing in
the given word-class.
\end_layout
\begin_layout Standard
It might be interesting to relate this to the size of the word-class.
\end_layout
\begin_layout Standard
\align center
\begin_inset Graphics
filename p5-run-11-merge/r11-cls-support.eps
width 50text%
\end_inset
\begin_inset Graphics
filename p5-run-12-merge/r12-cls-support.eps
width 50text%
\end_inset
\end_layout
\begin_layout Subsubsection*
Marginal Probability of the Class
\end_layout
\begin_layout Standard
The marginal probability of the class is as defined above:
\begin_inset Formula
\[
P\left(w\right)=P\left(w,*\right)=\sum_{d}P\left(w,d\right)
\]
\end_inset
The graph below shows
\begin_inset Formula $\log_{2}P\left(w\right)$
\end_inset
.
It might be interesting to look at how this correlates with the support.
\end_layout
\begin_layout Standard
\align center
\begin_inset Graphics
filename p5-run-11-merge/r11-cls-logli.eps
width 50text%
\end_inset
\begin_inset Graphics
filename p5-run-12-merge/r12-cls-logli.eps
width 50text%
\end_inset
\end_layout
\begin_layout Subsubsection*
Class Fractional Entropy
\end_layout
\begin_layout Standard
The entropy of a word is as defined earlier:
\begin_inset Formula
\[
h\left(w\right)=-\sum_{d}P\left(w,d\right)\log_{2}P\left(w,d\right)
\]
\end_inset
The fractional entropy is
\begin_inset Formula
\[
H\left(w\right)=\frac{h\left(w\right)}{P\left(w,*\right)}
\]
\end_inset
This is graphed below.
\end_layout
\begin_layout Standard
\align center
\begin_inset Graphics
filename p5-run-11-merge/r11-cls-fentropy.eps
width 50text%
\end_inset
\begin_inset Graphics
filename p5-run-12-merge/r12-cls-fentropy.eps
width 50text%
\end_inset
\end_layout
\begin_layout Subsubsection*
Compositional Entropy
\end_layout
\begin_layout Standard
The compositional entropy is a brand-new concept not discussed earlier in
this diary.
It is the entropy of the of the words that went in to form the class.
\end_layout
\begin_layout Standard
When a class is formed out of the words
\begin_inset Formula $w_{1},w_{2},\cdots,w_{n}$
\end_inset
, the total counts of each of these contributions
\begin_inset Formula $N\left(w_{1},*\right),N\left(w_{2},*\right),\cdots,N\left(w_{n},*\right)$
\end_inset
is tracked on the
\family sans
MemberLink
\family default
for that class.
Define the total observation count on this class as
\begin_inset Formula
\[
N_{C}=\sum_{k}N\left(w_{k},*\right)
\]
\end_inset
This can be used to define a class-relative fraction of how much each word
contributed to the class:
\begin_inset Formula
\[
P_{C}\left(w_{k},*\right)=\frac{N\left(w_{k},*\right)}{N_{C}}
\]
\end_inset
The compositional entropy is then the ordinary entropy with regards to this
compositional probability.
It is defined as
\begin_inset Formula
\[
CH=-\sum_{k}P_{C}\left(w_{k},*\right)\log_{2}P_{C}\left(w_{k},*\right)
\]
\end_inset
It is shown in the graph below.
\end_layout
\begin_layout Standard
\align center
\begin_inset Graphics
filename p5-run-11-merge/r11-cls-compo-entropy.eps
width 50text%
\end_inset
\begin_inset Graphics
filename p5-run-12-merge/r12-cls-compo-entropy.eps
width 50text%
\end_inset
\end_layout
\begin_layout Standard
The compositional entropy is interesting because it gives a hint of how
different words are contributing to form the class: it's a weighted average
of the contributions.
For example, for two words, the compositional entropy is maximized when
both words contribute equally (and is equal to 1.0).
Given how the above is scaling, it might be interesting to redo this graph,
dividing by (the log of) the number of words in the class.
\end_layout
\begin_layout Section*
Experiment-12
\end_layout
\begin_layout Standard
Some ongoing confusion motivates an attempt at a cleaner starting point
for merges.
Start with `
\family sans
run-1-t1234-tsup-1-1-1.rdb
\family default
` and build a new, clean file, using the scripts in `
\family sans
scm/gram-class/cleanup.scm
\family default
`.
Here's where it's at:
\end_layout
\begin_layout Standard
\begin_inset VSpace defskip
\end_inset
\end_layout
\begin_layout Standard
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="12" columns="4">
<features tabularvalignment="middle">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\family sans
r1-t1234-tsup-1-1-1
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\family sans
r12-mst.rdb
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\family sans
r12-shape.rdb
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $N_{L}$
\end_inset
= words
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
15083
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
9495
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
9495
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $N_{R}$
\end_inset
= dj
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
205003
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
204680
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
1015850
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $D_{\mbox{Tot}}$
\end_inset
= size
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
855718
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
833833
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
2717117
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $N_{\mbox{Tot}}$
\end_inset
= obs
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
7313131
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
7202301
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
22643824
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $\log_{2}N_{\mbox{word}}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
13.881
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
13.213
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
13.213
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $\log_{2}N_{\mbox{dj}}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
17.645
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
17.643
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
19.954
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $\log_{2}D_{\mbox{Tot}}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
19.707
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
19.669
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
21.374
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
sparsity
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
11.819
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
11.187
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
11.794
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
rarity
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
3.944
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
4.241
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
4.790
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $\log_{2}N_{\mbox{Tot}}/D_{\mbox{Tot}}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
3.095
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
3.111
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
3.059
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Entropy
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
16.352
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
15.720
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
18.117
\end_layout
\end_inset
</cell>
</row>
</lyxtabular>
\end_inset
\begin_inset VSpace defskip
\end_inset
\end_layout
\begin_layout Standard
Recall that
\begin_inset Formula
\[
\mbox{Sparsity}=-\log_{2}D_{\mbox{Tot}}/N_{L}\times N_{R}
\]
\end_inset
while
\begin_inset Formula
\[
\mbox{Rarity}=\log_{2}D_{\mbox{Tot}}/\sqrt{N_{L}\times N_{R}}
\]
\end_inset
\end_layout
\begin_layout Standard
Here's the word-disjunct entropies and MI's; its in r12-mi.rdb which is just
\family sans
`r12-shape.rdb
\family default
` with the MI scores in it.
\end_layout
\begin_layout Standard
\begin_inset VSpace defskip
\end_inset
\end_layout
\begin_layout Standard
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="2" columns="5">
<features tabularvalignment="middle">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
name
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Entropy total
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Left
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Right
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
MI
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
r12-mi.rdb
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
19.421
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
16.501
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
8.4683
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
5.5480
\end_layout
\end_inset
</cell>
</row>
</lyxtabular>
\end_inset
\begin_inset VSpace defskip
\end_inset
\end_layout
\begin_layout Standard
What's done:
\end_layout
\begin_layout Itemize
`
\family sans
r12-mst.rdb
\family default
` copies `
\family sans
run-1-t1234-tsup-1-1-1.rdb
\family default
` and forces bi-directional linkage symmetry.
It also recomputes support marginals and MM^T via contents of `
\family sans
run-common/marginals-mst.scm
\family default
`.
\end_layout
\begin_layout Itemize
`
\family sans
r12-shape.rdb
\family default
` adds shapes, and recomputed the MMT marginals for these.
The dimensions includes the CrossSections.
\end_layout
\begin_layout Itemize
`
\family sans
r12-mi.rdb
\family default
` batch-computes the word-disjunct MI (with `
\family sans
batch-all-pair-mi
\family default
`).
\end_layout
\begin_layout Itemize
`
\family sans
r12-mi-sim200.rdb
\family default
` is just `
\family sans
r12-mi.rdb
\family default
` with word-similarities pre-computed for the top 200 ranked words.
\end_layout
\begin_layout Subsection*
The Crash
\end_layout
\begin_layout Standard
As noted above, the runs crashed.
Will try restarting one more time, after nuking invalid CrossSections.
Nope.
That doesn't work.
\end_layout
\begin_layout Subsection*
Similarity between WordClasses vs.
Words
\end_layout
\begin_layout Standard
Two datasets halted at almost the same place.
Perhaps they are comparable.
As below.
\end_layout
\begin_layout Standard
\begin_inset VSpace defskip
\end_inset
\end_layout
\begin_layout Standard
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="3" columns="4">
<features tabularvalignment="middle">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Dataset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
nmerges
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
nclasses
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
nwords
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\family sans
r12-log-q0.7-c0.2-n3.dat
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
1062
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
536
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
9490
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\family sans
r12-log-q0.7-c0.2-n4.dat
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
1058
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
530
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
9486
\end_layout
\end_inset
</cell>
</row>
</lyxtabular>
\end_inset
\begin_inset VSpace defskip
\end_inset
\end_layout
\begin_layout Standard
In both cases, the number of word-classes that survived is almost exactly
half the number of merges run (why half?) This is because classes get repeatedl
y merged into, instead of having new classes being created out of the blue.
Kind of surprising, this needs to be understood better.
\begin_inset Foot
status collapsed
\begin_layout Plain Layout
The datasets for the below are located in the p5-class-sim directory.
The scripts used to build the datasets are in the utils/similarity-p5.scm.
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Anyway: One hypothesis: the similarity distributions between word-classes
should be low.
That should be quite distinct from one another, one might hope.
So, three batches of similarity distributions:
\end_layout
\begin_layout Itemize
Similarity between all of the 530 (or 536) word-classes.
(not counting self-similarity).
\end_layout
\begin_layout Itemize
Similarity between the top-ranked 530 words.
\end_layout
\begin_layout Itemize
Similarity between the top-ranked words and word-classes.
\end_layout
\begin_layout Standard
What will it be like? Can we expect the third bullet to show higher similarity
than the second?
\end_layout
\begin_layout Standard
All similarities will be recomputed from scratch, as the existing similarities
are tainted by the altered disjuncts – i.e.
the old similarities might not have been recomputed, after a change to
their disjuncts.
Only the
\begin_inset Quotes eld
\end_inset
precise
\begin_inset Quotes erd
\end_inset
merges recompute all similarities.
\end_layout
\begin_layout Subsubsection*
Conclusions – Summary
\end_layout
\begin_layout Standard
Lets summarize what was found below.
First, the self-similarity distribution for word-classes and other words
are dramatically different.
Word-classes have a high self-similarity.
The shapes of the distributions are different, too.
\end_layout
\begin_layout Standard
The distribution of non-zero similarities between classes, and classes and
words, and just words, are all more-or-less normal distributions (Gaussians),
all quite similar, all centered on a small negative MI (around -3 to -1.5,
depending) and fairly narrow (a standard deviation of about 3).
There are no dramatic differences at this level.
\end_layout
\begin_layout Standard
There is a very dramatic difference, when looking at the fraction of all
possible pairs with zero similarity! Zero similarity corresponds to
\begin_inset Formula $-\infty$
\end_inset
in the MI value; that is, there is zero overlap between the disjuncts.
They are orthogonal.
\end_layout
\begin_layout Standard
Of all of the possible similarities between the word-classes, only 7% are
non-zero.
The classes are highly orthogonal to one-another; only a few of them are
not.
By comparison, more than 80% of the words have non-zero similarity between
one another.
The words are mostly not orthogonal to one-another.
In the middle are the similarities between words and classes, coming in
at just 20% of all pairings.
\end_layout
\begin_layout Standard
Conclude: the classification algorithms is fairly good at
\begin_inset Quotes eld
\end_inset
orthogonalizing
\begin_inset Quotes erd
\end_inset
.
Post-classification, the classes are fairly distinct from one-another,
quite unlike the situation with raw words.
During classification, it appears that the algo is
\begin_inset Quotes eld
\end_inset
sweeping up
\begin_inset Quotes erd
\end_inset
words in an effective way: The 20% figure seems to say that classification
really is vacuuming up the words.
\end_layout
\begin_layout Standard
The above suggests a strong signal for the quality of classification: the
orthogonal fraction of pairings.
All three of the above figures should be tracked during classification.
\end_layout
\begin_layout Subsubsection*
Self-similarity Distributions
\end_layout
\begin_layout Standard
The similarity is given by the (symmetric) word-MI.
The self-similarity is the self-MI.
Here we go:
\end_layout
\begin_layout Standard
\align center
\begin_inset Graphics
filename p5-class-sim/self-mi.eps
width 80col%
\end_inset
\end_layout
\begin_layout Standard
Above is the distribution of the self-MI for the 530 word-classes and the
9490 words in
\family sans
r12-log-q0.7-c0.2-n3.dat.
\family default
Also, the 536 classes and 9486 words in
\family sans
r12-log-q0.7-c0.2-n4.dat.
\family default
Clearly, seems to be quite different.
What does it mean?
\end_layout
\begin_layout Subsubsection*
Class–Class Similarity Distributions
\end_layout
\begin_layout Standard
The noise=3 dataset had only 10154 non-vanishing similarities out of a total
of
\begin_inset Formula $(530\times531)/2=140715$
\end_inset
possible pairings of word-classes with one-another (excluding self-similarity).
For noise=4, there were only 9988 out a possible 536 x 537/2 = 143916 pairings.
This means that only seven percent (approx.) of the word-classes are similar
to each other enough to have at least one disjunct in common.
The rest had zero disjuncts in common! So, word classes are not perfectly
orthogonal to one-another, but most of them are, considered pair-wise.
That's a pretty hot result.
I like it.
\end_layout
\begin_layout Standard
\align center
\begin_inset Graphics
filename p5-class-sim/class-mi.eps
width 80col%
\end_inset
\end_layout
\begin_layout Standard
The Gaussian G(-3,3.5) is just an eyeballed fit, as usual.
The noise=3 curve seems to be shifted slightly to the left.
One would expect that better clustering shifts these curves leftward.
\end_layout
\begin_layout Subsubsection*
Word–Word Similarity Distributions
\end_layout
\begin_layout Standard
The noise=3 dataset had 117218 non-vanishing similarities out of a total
of
\begin_inset Formula $(530\times531)/2=140715$
\end_inset
possible pairings of word-classes with one-another (excluding self-similarity).
For noise=4, there were 117023 out a possible 536 x 537/2 = 143916 pairings.
So this time, approx 83% of all possible similarity pairs have similarities.
A very non-orthogonal situation.
\end_layout
\begin_layout Standard
The selected words were the top-ranked ones, out of all possible words.
The number of words was limited, with the idea that they would be comparable
in number to the classes.
\end_layout
\begin_layout Standard
\align center
\begin_inset Graphics
filename p5-class-sim/word-mi.eps
width 80col%
\end_inset
\end_layout
\begin_layout Standard
The Gaussian G(-1.5,3.1) is just an eyeballed fit, as usual.
Note that it is shifted to the right, and is narrower, than the Gaussian
in the previous figure.
The noise=3 curve seems to be shifted slightly to the left.
It is not clear why it should be shifted.
\end_layout
\begin_layout Standard
Compare this to the earlier, pre-merge word-word MI distribution graph,
about half-way up in this document.
It's the same distribution, but before any merges have been done.
This is shown below, relative to the current noise=3 graph.
Both the mean and the stddev are about the same; maybe a bit of a shift
of the mean.
\end_layout
\begin_layout Standard
\align center
\begin_inset Graphics
filename p5-class-sim/pre-post.eps
width 80col%
\end_inset
\end_layout
\begin_layout Standard
The
\begin_inset Quotes eld
\end_inset
pre-merge
\begin_inset Quotes erd
\end_inset
curve is the one from half-way up: it shows similarities for only 200 words,
and so is noisier.
The shape of the fat tail is ambiguous.
Earlier, it looked like a well-defined Gaussian with a very fat tail; but
here, this is not so clear.
Clearly, noise=4 suppresses the tail a lot more than noise=3.
\end_layout
\begin_layout Subsubsection*
Class–Word Similarity Distributions
\end_layout
\begin_layout Standard
The noise=3 dataset had 58555 non-vanishing similarities out of a total
of
\begin_inset Formula $530^{2}=280900$
\end_inset
possible pairings of word-classes with one-another (excluding self-similarity).
For noise=4, there were 55194 out a possible
\begin_inset Formula $536^{2}=287296$
\end_inset
pairings.
So this is a relatively meek 20% of all possible pairings.
Conclude that the word-classes have swept up those things that have overlaps.
\end_layout
\begin_layout Standard
\align center
\begin_inset Graphics
filename p5-class-sim/clawrd-mi.eps
width 80col%
\end_inset
\end_layout
\begin_layout Standard
The Gaussian G(-1.5,3.1) is just an eyeballed fit, as usual.
This is the same Gaussian as used above.
The noise=3 curve seems to be shifted slightly to the left.
It is not clear why it should be shifted.
\end_layout
\begin_layout Subsubsection*
All-in-one Similarity Distributions
\end_layout
\begin_layout Standard
The below reproduces all of the above curves, in one figure.
\end_layout
\begin_layout Standard
\align center
\begin_inset Graphics
filename p5-class-sim/all-mi.eps
width 80col%
\end_inset
\end_layout
\begin_layout Standard
Hmm.
Graphed this way, it seems that there are no dramatic differences.
There are subtle differences, already noted.
\end_layout
\begin_layout Standard
To conclude: perhaps the most surprising is not the distributions of the
similarities, but the fraction of the pairs that have non-zero similarity.
This is the dramatic result we were looking for, even if its not evident
in these graphs.
\end_layout
\begin_layout Section*
The End
\end_layout
\begin_layout Standard
This is the end of Part Five of the diary.
\end_layout
\end_body
\end_document