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Question: Define entropy and explain types of entropy

Mumbai University >Information Technology> Sem 4 > Information Theory & Coding

Marks: 4 Marks

Year: May 2016

modified 6 weeks ago  • written 8 weeks ago by gravatar for Veena Nandi Veena Nandi70

The most fundamental concept of information theory is the entropy. The entropy is defined as average amount of information per message. The entropy of a random variable X is defined by,

H(X) =-Σx P(x) log p(x)

H(X)≥ 0, entropy is always non-negative. H(X)=0 if X is deterministic.

  • Since Hb(X) = logb(a)Ha(X), we don‟t need to specify the base of the logarithm.
  • The entropy is non-negative. It is zero when the random variable is “certain” to be predicted. Entropy is defined using the Clausius inequality.
  • Entropy is defined in terms of probabilistic behavior of a source of information. In information theory the source output are discrete random variables that have a certain fixed finite alphabet with certain probabilities. Entropy is average information content for the given source symbol.
  • Entropy (example): Binary memory less source has symbols 0 and 1 which have probabilities p0 and p1 (1-p0). Count the entropy as a function of p0.
  • Entropy is measured in bits (the log is log2)

There are two types of Entropy:

  1. Joint Entropy
  2. Conditional Entropy

Joint Entropy:

Joint entropy is entropy of joint probability distribution, or a multi valued random variables. If X and Y are discrete random variables and f(x, y) is the value of their joint probability distribution of (x, y), then the joint entropy of X and Y is

H(X, Y)=-Σ x є X Σ y є Y f(x, y) log f(x, y)

The joint entropy represents the amount of information needed on average to specify the value of two discrete random variables.

Conditional Entropy:

Average condition self-information is called as the condition entropy. If X and Y are discrete random variables and f(x, y) and f (y |x) are the values of their joints and conditional probability distributions, then:

H (Y|X) =-ΣxєXΣyє Y f(x, y) log f(y |x) is the conditional entropy of Y given X.

  • The conditional entropy indicates how much extra information you still need to supply on average to communicate Y given that other party knows X.
  • Conditional Entropy is also called as equivocation. Conditional entropy is the amount of information in one random variable given we already know the other.
written 8 weeks ago by gravatar for Veena Nandi Veena Nandi70
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