Mixing
Prove an inequality concerning Kullback-Leibler Divergence
$begingroup$
For any distribution $P$ and $Q$ on $mathcal{X}$ and any function $f:mathcal{X} rightarrow mathbb{R}$, prove the following inequality:
$$mathbb{E}_{xsim Q}[f(x)]le ln mathbb{E}_{xsim P}[exp(f(x))]+KL(Q||P)$$
I have no idea on transforming the expectation to a Kullback-Leibler Divergence at all. Is there a simple proof on the inequality (for example, just using the knowledge of probability theory and calculas)? Thank you!
probability-theory machine-learning information-theory
asked Jan 11 at 6:36
zbh2047zbh2047
356
$endgroup$
add a comment |
$begingroup$
For any distribution $P$ and $Q$ on $mathcal{X}$ and any function $f:mathcal{X} rightarrow mathbb{R}$, prove the following inequality:
$$mathbb{E}_{xsim Q}[f(x)]le ln mathbb{E}_{xsim P}[exp(f(x))]+KL(Q||P)$$
I have no idea on transforming the expectation to a Kullback-Leibler Divergence at all. Is there a simple proof on the inequality (for example, just using the knowledge of probability theory and calculas)? Thank you!
probability-theory machine-learning information-theory
asked Jan 11 at 6:36
zbh2047zbh2047
356
$endgroup$
add a comment |
$begingroup$
For any distribution $P$ and $Q$ on $mathcal{X}$ and any function $f:mathcal{X} rightarrow mathbb{R}$, prove the following inequality:
$$mathbb{E}_{xsim Q}[f(x)]le ln mathbb{E}_{xsim P}[exp(f(x))]+KL(Q||P)$$
I have no idea on transforming the expectation to a Kullback-Leibler Divergence at all. Is there a simple proof on the inequality (for example, just using the knowledge of probability theory and calculas)? Thank you!
probability-theory machine-learning information-theory
asked Jan 11 at 6:36
zbh2047zbh2047
356
$endgroup$
For any distribution $P$ and $Q$ on $mathcal{X}$ and any function $f:mathcal{X} rightarrow mathbb{R}$, prove the following inequality:
$$mathbb{E}_{xsim Q}[f(x)]le ln mathbb{E}_{xsim P}[exp(f(x))]+KL(Q||P)$$
I have no idea on transforming the expectation to a Kullback-Leibler Divergence at all. Is there a simple proof on the inequality (for example, just using the knowledge of probability theory and calculas)? Thank you!
probability-theory machine-learning information-theory
probability-theory machine-learning information-theory
asked Jan 11 at 6:36
zbh2047zbh2047
356
asked Jan 11 at 6:36
zbh2047zbh2047
356
asked Jan 11 at 6:36
zbh2047zbh2047
356
asked Jan 11 at 6:36
zbh2047zbh2047
356
asked Jan 11 at 6:36
zbh2047zbh2047
356
356
add a comment |
add a comment |
1 Answer
1
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oldest
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$begingroup$
This computation appears in the study of variational inference. Note that
$$
begin{align}
log mathbb{E}_{xsim P}(f(x)) &triangleq log int_x p(x) f(x) dx\
&= log int_x q(x) frac{p(x)}{q(x)}f(x) dx\
&geq int_x q(x) log left(frac{p(x)}{q(x)}f(x) right)dx\
&= int_x q(x) log f(x) dx +int_x q(x) log left(frac{p(x)}{q(x)} right)dx\
&triangleq mathbb{E}_{xsim Q}(log(f(x)))-KL(Q||P),
end{align}
$$
answered Jan 11 at 7:30
SteliosStelios
2,141279
$endgroup$
$begingroup$
To note, the inequality is due to Jensen's, with concavity of $log(cdot)$.
$endgroup$
– Aaron
Jan 11 at 15:53
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
This computation appears in the study of variational inference. Note that
$$
begin{align}
log mathbb{E}_{xsim P}(f(x)) &triangleq log int_x p(x) f(x) dx\
&= log int_x q(x) frac{p(x)}{q(x)}f(x) dx\
&geq int_x q(x) log left(frac{p(x)}{q(x)}f(x) right)dx\
&= int_x q(x) log f(x) dx +int_x q(x) log left(frac{p(x)}{q(x)} right)dx\
&triangleq mathbb{E}_{xsim Q}(log(f(x)))-KL(Q||P),
end{align}
$$
answered Jan 11 at 7:30
SteliosStelios
2,141279
$endgroup$
$begingroup$
To note, the inequality is due to Jensen's, with concavity of $log(cdot)$.
$endgroup$
– Aaron
Jan 11 at 15:53
add a comment |
$begingroup$
This computation appears in the study of variational inference. Note that
$$
begin{align}
log mathbb{E}_{xsim P}(f(x)) &triangleq log int_x p(x) f(x) dx\
&= log int_x q(x) frac{p(x)}{q(x)}f(x) dx\
&geq int_x q(x) log left(frac{p(x)}{q(x)}f(x) right)dx\
&= int_x q(x) log f(x) dx +int_x q(x) log left(frac{p(x)}{q(x)} right)dx\
&triangleq mathbb{E}_{xsim Q}(log(f(x)))-KL(Q||P),
end{align}
$$
answered Jan 11 at 7:30
SteliosStelios
2,141279
$endgroup$
$begingroup$
To note, the inequality is due to Jensen's, with concavity of $log(cdot)$.
$endgroup$
– Aaron
Jan 11 at 15:53
add a comment |
$begingroup$
This computation appears in the study of variational inference. Note that
$$
begin{align}
log mathbb{E}_{xsim P}(f(x)) &triangleq log int_x p(x) f(x) dx\
&= log int_x q(x) frac{p(x)}{q(x)}f(x) dx\
&geq int_x q(x) log left(frac{p(x)}{q(x)}f(x) right)dx\
&= int_x q(x) log f(x) dx +int_x q(x) log left(frac{p(x)}{q(x)} right)dx\
&triangleq mathbb{E}_{xsim Q}(log(f(x)))-KL(Q||P),
end{align}
$$
answered Jan 11 at 7:30
SteliosStelios
2,141279
$endgroup$
This computation appears in the study of variational inference. Note that
$$
begin{align}
log mathbb{E}_{xsim P}(f(x)) &triangleq log int_x p(x) f(x) dx\
&= log int_x q(x) frac{p(x)}{q(x)}f(x) dx\
&geq int_x q(x) log left(frac{p(x)}{q(x)}f(x) right)dx\
&= int_x q(x) log f(x) dx +int_x q(x) log left(frac{p(x)}{q(x)} right)dx\
&triangleq mathbb{E}_{xsim Q}(log(f(x)))-KL(Q||P),
end{align}
$$
answered Jan 11 at 7:30
SteliosStelios
2,141279
answered Jan 11 at 7:30
SteliosStelios
2,141279
answered Jan 11 at 7:30
SteliosStelios
2,141279
answered Jan 11 at 7:30
SteliosStelios
2,141279
2,141279
$begingroup$
To note, the inequality is due to Jensen's, with concavity of $log(cdot)$.
$endgroup$
– Aaron
Jan 11 at 15:53
add a comment |
$begingroup$
To note, the inequality is due to Jensen's, with concavity of $log(cdot)$.
$endgroup$
– Aaron
Jan 11 at 15:53
$begingroup$
To note, the inequality is due to Jensen's, with concavity of $log(cdot)$.
$endgroup$
– Aaron
Jan 11 at 15:53
$begingroup$
To note, the inequality is due to Jensen's, with concavity of $log(cdot)$.
$endgroup$
– Aaron
Jan 11 at 15:53
add a comment |
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