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Suppose we have an expression

f(x, h(x,y)), for some function f and h, and x, y are random variables,

now we know that the function f(a, b) is concave w.r.t. a for given b. Can we use Jensen's inequality to obtain

$E[f(x, h(x,y))] < E_{x,y}[f(E[x], h(x,y))]$

where $E_{x,y}$ means expectation over (x,y).

Thanks.

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  • $\begingroup$ Let me make it clearer. It should be expectation over x and y. $\endgroup$
    – Michael Fan Zhang
    Commented Dec 10, 2014 at 10:50
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    $\begingroup$ It is trivially true if the inner expectation is a conditional expectation of $x$, given $h(x,y)$, otherwise I don't see why it should be true... $\endgroup$
    – Martin Hairer
    Commented Dec 10, 2014 at 13:58
  • $\begingroup$ so u mean the RHS of the above inequality is $E[f(E[x|h(x,y)],h(x,y))]$? $\endgroup$
    – Michael Fan Zhang
    Commented Dec 11, 2014 at 2:12

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It's not true for h(x,y)=x, f(a,b)=-(a-b)^2 .

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