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Concavity of entropy

  1. Jul 23, 2009 #1
    1. The problem statement, all variables and given/known data
    Shannon entropy is a concave function defined as follows:
    [tex]H(X)=-\sum_{x}p(x)\log p(x)[/tex]

    Conditional Shannon entropy is defined as follows:
    [tex]H(X|Y)=\sum_{y} p(y) H(X|Y=y)=-\sum_{y} p(y)\sum_{x}p(x|Y=y)\log p(x|Y=y)[/tex]

    Can we deduce that:
    [tex]\sum_{y} p(y)H(X|Y=y)\geq H(X|Y=y)[/tex]


    2. Relevant equations



    3. The attempt at a solution
    I would say yes because of the concavity but I am confused with the 2 random variables.
     
  2. jcsd
  3. Jul 24, 2009 #2

    jmb

    User Avatar

    I'm assuming the RHS of your final expression is also meant to be summed over [tex]y[/tex], otherwise it doesn't make much sense...

    The way to go about this is to think about the nature of [tex]p(y)[/tex]. What constraints do you know about the values that [tex]p(y)[/tex] can take?

    PS Do you really mean [tex]\geq[/tex] in the last line, or do you mean [tex]\leq[/tex]?
     
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