Difference between revisions of "Entropy, Relative Entropy, Mutual Information"

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=== Expected Value ===
 
=== Expected Value ===
Given a random variable, <math>X</math> with probability mass function <math>p\left(x\right)</math>, the expected value of <math>g\left(X\right)</math> is
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For a discrete random variable, <math>X</math> with probability mass function <math>p\left(x\right)</math>, the expected value of <math>g\left(X\right)</math> is
  
 
{{NumBlk|:|<math>E\left[g\left(X\right)\right]=\sum_{x\in \mathcal{X}} g\left(x\right)\cdot p\left(x\right)</math>|{{EquationRef|2}}}}
 
{{NumBlk|:|<math>E\left[g\left(X\right)\right]=\sum_{x\in \mathcal{X}} g\left(x\right)\cdot p\left(x\right)</math>|{{EquationRef|2}}}}

Revision as of 15:38, 25 June 2020

Definitions

Entropy

  • a measure of the uncertainty of a random variable
  • The entropy of a random variable is a measure of the uncertainty of the random variable
    • it is a measure of the amount of information required on the average to describe the random variable

Relative Entropy

  • a measure of the distance between two distributions
  • a measure of the inefficiency of assuming that the distribution is when the true distribution is .

Mutual Information

  • a measure of the amount of information that one random variable contains about another random variable

Entropy

The entropy of a discrete random variable, , is

 

 

 

 

(1)

where has a probability mass function (pmf), , and an alphabet .

Expected Value

For a discrete random variable, with probability mass function , the expected value of is

 

 

 

 

(2)