Simple Expression Related to Mutual Information












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One way to define the mutual information is



$I(X;Y) = H(X) - H(X|Y)$



I have found it useful to look the related quantity



$?(X;Y=y) = H(X) - H(X|Y=y)$



That is, we look at how much the entropy of $X$ is decreased given a particular outcome $y$ for $Y$.



It is not hard to see that the mutual information is regained on expectation on $Y$, so that we have



$E_Y[?(X;Y=y)] \
= sum_y p(y)(H(X) - H(X|Y=y)) \
= H(X) - sum_y p(y)H(X|Y=y) \
= H(X) - H(X|Y) \
= I(X;Y)$



My question: does my $?(X;Y=y)$ function have a name? Or a standardized notation?



Note it's not the same as pointwise mutual information: rather, I think this would be the expectation of pointwise mutual information, but only on $X$ (rather than both variables). So it's "in between" the regular mutual information and the pointwise version.










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    0














    One way to define the mutual information is



    $I(X;Y) = H(X) - H(X|Y)$



    I have found it useful to look the related quantity



    $?(X;Y=y) = H(X) - H(X|Y=y)$



    That is, we look at how much the entropy of $X$ is decreased given a particular outcome $y$ for $Y$.



    It is not hard to see that the mutual information is regained on expectation on $Y$, so that we have



    $E_Y[?(X;Y=y)] \
    = sum_y p(y)(H(X) - H(X|Y=y)) \
    = H(X) - sum_y p(y)H(X|Y=y) \
    = H(X) - H(X|Y) \
    = I(X;Y)$



    My question: does my $?(X;Y=y)$ function have a name? Or a standardized notation?



    Note it's not the same as pointwise mutual information: rather, I think this would be the expectation of pointwise mutual information, but only on $X$ (rather than both variables). So it's "in between" the regular mutual information and the pointwise version.










    share|cite|improve this question

























      0












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      1





      One way to define the mutual information is



      $I(X;Y) = H(X) - H(X|Y)$



      I have found it useful to look the related quantity



      $?(X;Y=y) = H(X) - H(X|Y=y)$



      That is, we look at how much the entropy of $X$ is decreased given a particular outcome $y$ for $Y$.



      It is not hard to see that the mutual information is regained on expectation on $Y$, so that we have



      $E_Y[?(X;Y=y)] \
      = sum_y p(y)(H(X) - H(X|Y=y)) \
      = H(X) - sum_y p(y)H(X|Y=y) \
      = H(X) - H(X|Y) \
      = I(X;Y)$



      My question: does my $?(X;Y=y)$ function have a name? Or a standardized notation?



      Note it's not the same as pointwise mutual information: rather, I think this would be the expectation of pointwise mutual information, but only on $X$ (rather than both variables). So it's "in between" the regular mutual information and the pointwise version.










      share|cite|improve this question













      One way to define the mutual information is



      $I(X;Y) = H(X) - H(X|Y)$



      I have found it useful to look the related quantity



      $?(X;Y=y) = H(X) - H(X|Y=y)$



      That is, we look at how much the entropy of $X$ is decreased given a particular outcome $y$ for $Y$.



      It is not hard to see that the mutual information is regained on expectation on $Y$, so that we have



      $E_Y[?(X;Y=y)] \
      = sum_y p(y)(H(X) - H(X|Y=y)) \
      = H(X) - sum_y p(y)H(X|Y=y) \
      = H(X) - H(X|Y) \
      = I(X;Y)$



      My question: does my $?(X;Y=y)$ function have a name? Or a standardized notation?



      Note it's not the same as pointwise mutual information: rather, I think this would be the expectation of pointwise mutual information, but only on $X$ (rather than both variables). So it's "in between" the regular mutual information and the pointwise version.







      statistics information-theory entropy






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      asked yesterday









      Mike Battaglia

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      1,2581126






















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