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Concept
Conditional entropy
Summary
In information theory, the conditional entropy quantifies the amount of information needed to describe the outcome of a random variable given that the value of another random variable is known. Here, information is measured in shannons, nats, or hartleys. The entropy of conditioned on is written as .
The conditional entropy of given is defined as
where and denote the support sets of and .
Note: Here, the convention is that the expression should be treated as being equal to zero. This is because .
Intuitively, notice that by definition of expected value and of conditional probability, can be written as , where is defined as . One can think of as associating each pair with a quantity measuring the information content of given . This quantity is directly related to the amount of information needed to describe the event given . Hence by computing the expected value of over all pairs of values , the conditional entropy measures how much information, on average, the variable encodes about .
Let be the entropy of the discrete random variable conditioned on the discrete random variable taking a certain value . Denote the support sets of and by and . Let have probability mass function . The unconditional entropy of is calculated as , i.e.
where is the information content of the outcome of taking the value . The entropy of conditioned on taking the value is defined analogously by conditional expectation:
Note that is the result of averaging over all possible values that may take. Also, if the above sum is taken over a sample , the expected value is known in some domains as equivocation.
Given discrete random variables with image and with image , the conditional entropy of given is defined as the weighted sum of for each possible value of , using as the weights:
if and only if the value of is completely determined by the value of .
Conversely, if and only if and are independent random variables.
Assume that the combined system determined by two random variables and has joint entropy , that is, we need bits of information on average to describe its exact state.
Official source
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