Typical set vs. smallest delta-sufficient subset

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Simba

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Feb 25, 2009, 8:26:20 AM2/25/09
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Hi,
I studied the source coding theorem, chapter 4 of the MacKay's book,
but I have some perplexities.
The source coding theorem implies that, for large N, the cardinality
of the smallest delta-sufficient subset is about 2^(NH). So, in this
set we have the 2^(NH) most probable outcomes.
There is also the asymptotic equipartition principle, which implies
that, for large N, the typical set, whose elements have probability of
'about' 2^(-NH), contains almost all the probability. So, the typical
set has about 2^(NH) elements.
So, we have that the typical set and the smallest delta-sufficient
subset for large N have approximately the same cardinality, 2^(NH),
right?
But the typical set doesn't contain the most probable outcomes (as for
the other set), so I imagine it as the smallest delta-sufficient
subset "shifted" a bit towards the less probable outcomes, where
"shifted" refers to a picture in which the outcomes are represented as
points on segment, with probability increasing from left to right: the
smallest delta-sufficient subset is at the extreme right, while the
typical set is a bit more centered, right?
So, the book says that we can (and probably should) define a
compression algorithm that gives a distinct name of length NH bits to
each element of the typical set. That's ok, but my question is: why
the typical set and not the smallest delta-sufficient set? I know that
it should be equivalent, because both sets contains almost all the
probability, but why should we choose the typical set?
Thanks

Frederico Guth

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May 25, 2021, 5:10:57 PMMay 25
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I am studying this myself. Here is my take. A Typical subset is a delta-sufficient subset. But not all delta-sufficient subsets are typical. Why? Because the elements of the typical subset have similar probabilities, while the smallest delta-sufficient subset, for example, is not typical (by construction. You chose the most probable elements).


Why to compress with the typical and not with the smallest delta-sufficient subset:
Imagine a very naive compression of a text. You may choose to compress using the smallest delta-sufficient subset of the letters of the alphabet. So, e is the most probable letter and you will keep it. q is the least used letter and it is not in your subset. The problem is that the least used letters are more informative than the most used letters. You can probabily infr a txt without the lttr 'e'. The typical set is the most informative subset of your alphabet.

Fred

Eli the Bearded

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May 25, 2021, 5:56:24 PMMay 25
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In comp.compression, Frederico Guth <fred...@fredguth.com> wrote:
> I am studying this myself. Here is my take. A Typical subset is a
> delta-sufficient subset. But not all delta-sufficient subsets are
> typical. Why? Because the elements of the typical subset have similar
> probabilities, while the smallest delta-sufficient subset, for example,
> is not typical (by construction. You chose the most probable elements).
...
> On Wednesday, February 25, 2009 at 10:26:20 AM UTC-3, Simba wrote:
>> Hi,
>> I studied the source coding theorem, chapter 4 of the MacKay's book,
>> but I have some perplexities.

Top-posted reply 12 years after the original post.

By construction, the least useful subset of replies are those that are
composed long after an answer is expected.

Elijah
------
need to wait a few more decades to for the smallest possible usefulness
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