Expectation of log-density with respect to a different random variable
$begingroup$
Let $X$ and $Y$ be two continuous random variables with marginal density functions $f_{X}(x) $ and $f_{Y}(y) $.
Is it true that $$ E[log f_{X}(X)] geq E[log f_{Y}(X)] ?$$
Perhaps the concavity of the log function will come in handy, along with Jensen's inequality, but I am having particular trouble manipulating the $ f_{Y}(X) $ term.
probability-theory inequality expectation
edited Jan 21 '18 at 23:26
zoli
17.1k41945
asked Jan 21 '18 at 15:26
E WernerE Werner
655
$endgroup$
add a comment |
$begingroup$
Let $X$ and $Y$ be two continuous random variables with marginal density functions $f_{X}(x) $ and $f_{Y}(y) $.
Is it true that $$ E[log f_{X}(X)] geq E[log f_{Y}(X)] ?$$
Perhaps the concavity of the log function will come in handy, along with Jensen's inequality, but I am having particular trouble manipulating the $ f_{Y}(X) $ term.
probability-theory inequality expectation
edited Jan 21 '18 at 23:26
zoli
17.1k41945
asked Jan 21 '18 at 15:26
E WernerE Werner
655
$endgroup$
add a comment |
$begingroup$
Let $X$ and $Y$ be two continuous random variables with marginal density functions $f_{X}(x) $ and $f_{Y}(y) $.
Is it true that $$ E[log f_{X}(X)] geq E[log f_{Y}(X)] ?$$
Perhaps the concavity of the log function will come in handy, along with Jensen's inequality, but I am having particular trouble manipulating the $ f_{Y}(X) $ term.
probability-theory inequality expectation
edited Jan 21 '18 at 23:26
zoli
17.1k41945
asked Jan 21 '18 at 15:26
E WernerE Werner
655
$endgroup$
Let $X$ and $Y$ be two continuous random variables with marginal density functions $f_{X}(x) $ and $f_{Y}(y) $.
Is it true that $$ E[log f_{X}(X)] geq E[log f_{Y}(X)] ?$$
Perhaps the concavity of the log function will come in handy, along with Jensen's inequality, but I am having particular trouble manipulating the $ f_{Y}(X) $ term.
probability-theory inequality expectation
probability-theory inequality expectation
edited Jan 21 '18 at 23:26
zoli
17.1k41945
asked Jan 21 '18 at 15:26
E WernerE Werner
655
edited Jan 21 '18 at 23:26
zoli
17.1k41945
asked Jan 21 '18 at 15:26
E WernerE Werner
655
edited Jan 21 '18 at 23:26
zoli
17.1k41945
edited Jan 21 '18 at 23:26
zoli
17.1k41945
edited Jan 21 '18 at 23:26
zoli
17.1k41945
17.1k41945
asked Jan 21 '18 at 15:26
E WernerE Werner
655
asked Jan 21 '18 at 15:26
E WernerE Werner
655
asked Jan 21 '18 at 15:26
E WernerE Werner
655
655
add a comment |
add a comment |
1 Answer
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$begingroup$
By Jensen's inequality,
$$
mathsf{E}log frac{f_Y(X)}{f_X(X)}le log mathsf{E}frac{f_Y(X)}{f_X(X)}=log int frac{f_Y(x)}{f_X(x)}f_X(x)dx=0
$$
answered Jan 2 at 3:20
d.k.o.d.k.o.
10.6k730
$endgroup$
add a comment |
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1 Answer
1
active
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1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
By Jensen's inequality,
$$
mathsf{E}log frac{f_Y(X)}{f_X(X)}le log mathsf{E}frac{f_Y(X)}{f_X(X)}=log int frac{f_Y(x)}{f_X(x)}f_X(x)dx=0
$$
answered Jan 2 at 3:20
d.k.o.d.k.o.
10.6k730
$endgroup$
add a comment |
$begingroup$
By Jensen's inequality,
$$
mathsf{E}log frac{f_Y(X)}{f_X(X)}le log mathsf{E}frac{f_Y(X)}{f_X(X)}=log int frac{f_Y(x)}{f_X(x)}f_X(x)dx=0
$$
answered Jan 2 at 3:20
d.k.o.d.k.o.
10.6k730
$endgroup$
add a comment |
$begingroup$
By Jensen's inequality,
$$
mathsf{E}log frac{f_Y(X)}{f_X(X)}le log mathsf{E}frac{f_Y(X)}{f_X(X)}=log int frac{f_Y(x)}{f_X(x)}f_X(x)dx=0
$$
answered Jan 2 at 3:20
d.k.o.d.k.o.
10.6k730
$endgroup$
By Jensen's inequality,
$$
mathsf{E}log frac{f_Y(X)}{f_X(X)}le log mathsf{E}frac{f_Y(X)}{f_X(X)}=log int frac{f_Y(x)}{f_X(x)}f_X(x)dx=0
$$
answered Jan 2 at 3:20
d.k.o.d.k.o.
10.6k730
answered Jan 2 at 3:20
d.k.o.d.k.o.
10.6k730
answered Jan 2 at 3:20
d.k.o.d.k.o.
10.6k730
answered Jan 2 at 3:20
d.k.o.d.k.o.
10.6k730
10.6k730
add a comment |
add a comment |
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