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March 26th, 2013, 11:04 PM   #1
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Entropy H is continuous in p(i)

Here's something that I came across in "A Mathematical theory of communication" By Shannon that I couldn't get-

Quote:
" Suppose we have a set of possible events whose probabilities of occurrence are p1, p2, ... pn. These probabilities are known but that is all we know concerning which event will occur. Can we find a measure of how much “choice” is involved in the selection of the event or of how uncertain we are of the outcome? If there is such a measure, say H(p1, p2,...pn), it is reasonable to require of it the following properties:

1. H should be continuous in the p(i). "
H is the entropy.


What does he mean by H should be continuous in p(i) [where p(i) indicates probability of 'i' th event]. I couldn't get it.
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March 27th, 2013, 05:16 AM   #2
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Re: Entropy H is continuous in p(i)

If p(i) changes by a very small amount, H will change by a small amount. You can use the normal epsilon-delta definition here if you prefer.
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