Upper bound for differences between two expectations
$begingroup$
$f : mathbb R^n rightarrow mathbb R$. Is there a good upper bound for the following difference?
begin{equation*}
big| mathbb E_{(x_1, ldots, x_n) sim nu} f(x_1, ldots, x_n)
- mathbb E_{(x_1, ldots, x_n) sim mu^n} f(x_1, ldots, x_n) big|
end{equation*}
the marginal probability distribution of $nu$ is $mu$
probability probability-distributions expected-value upper-lower-bounds marginal-distribution
asked Dec 27 '18 at 14:37
Rui ZhangRui Zhang
316
$endgroup$
add a comment |
$begingroup$
$f : mathbb R^n rightarrow mathbb R$. Is there a good upper bound for the following difference?
begin{equation*}
big| mathbb E_{(x_1, ldots, x_n) sim nu} f(x_1, ldots, x_n)
- mathbb E_{(x_1, ldots, x_n) sim mu^n} f(x_1, ldots, x_n) big|
end{equation*}
the marginal probability distribution of $nu$ is $mu$
probability probability-distributions expected-value upper-lower-bounds marginal-distribution
asked Dec 27 '18 at 14:37
Rui ZhangRui Zhang
316
$endgroup$
1
$begingroup$
Unfortunately the best bound is $(sup f) - (inf f)$. For example, for bounded $inf$ and $sup$, take $U, V$ iid uniform over $[0,1]$ and for distribution 1 consider $(X_1,Y_1)=(U,U)$, for distribution 2 consider $(X_2,Y_2)=(U,V)$. Then define $f(x,x)=a$ and $f(x,y) = b$ whenever $x neq y$. So the difference is $|a-b|$ which is arbitrarily large.
$endgroup$
– Michael
Dec 27 '18 at 15:03
add a comment |
$begingroup$
$f : mathbb R^n rightarrow mathbb R$. Is there a good upper bound for the following difference?
begin{equation*}
big| mathbb E_{(x_1, ldots, x_n) sim nu} f(x_1, ldots, x_n)
- mathbb E_{(x_1, ldots, x_n) sim mu^n} f(x_1, ldots, x_n) big|
end{equation*}
the marginal probability distribution of $nu$ is $mu$
probability probability-distributions expected-value upper-lower-bounds marginal-distribution
asked Dec 27 '18 at 14:37
Rui ZhangRui Zhang
316
$endgroup$
$f : mathbb R^n rightarrow mathbb R$. Is there a good upper bound for the following difference?
begin{equation*}
big| mathbb E_{(x_1, ldots, x_n) sim nu} f(x_1, ldots, x_n)
- mathbb E_{(x_1, ldots, x_n) sim mu^n} f(x_1, ldots, x_n) big|
end{equation*}
the marginal probability distribution of $nu$ is $mu$
probability probability-distributions expected-value upper-lower-bounds marginal-distribution
probability probability-distributions expected-value upper-lower-bounds marginal-distribution
asked Dec 27 '18 at 14:37
Rui ZhangRui Zhang
316
asked Dec 27 '18 at 14:37
Rui ZhangRui Zhang
316
asked Dec 27 '18 at 14:37
Rui ZhangRui Zhang
316
asked Dec 27 '18 at 14:37
Rui ZhangRui Zhang
316
asked Dec 27 '18 at 14:37
Rui ZhangRui Zhang
316
316
1
$begingroup$
Unfortunately the best bound is $(sup f) - (inf f)$. For example, for bounded $inf$ and $sup$, take $U, V$ iid uniform over $[0,1]$ and for distribution 1 consider $(X_1,Y_1)=(U,U)$, for distribution 2 consider $(X_2,Y_2)=(U,V)$. Then define $f(x,x)=a$ and $f(x,y) = b$ whenever $x neq y$. So the difference is $|a-b|$ which is arbitrarily large.
$endgroup$
– Michael
Dec 27 '18 at 15:03
add a comment |
1
$begingroup$
Unfortunately the best bound is $(sup f) - (inf f)$. For example, for bounded $inf$ and $sup$, take $U, V$ iid uniform over $[0,1]$ and for distribution 1 consider $(X_1,Y_1)=(U,U)$, for distribution 2 consider $(X_2,Y_2)=(U,V)$. Then define $f(x,x)=a$ and $f(x,y) = b$ whenever $x neq y$. So the difference is $|a-b|$ which is arbitrarily large.
$endgroup$
– Michael
Dec 27 '18 at 15:03
1
1
$begingroup$
Unfortunately the best bound is $(sup f) - (inf f)$. For example, for bounded $inf$ and $sup$, take $U, V$ iid uniform over $[0,1]$ and for distribution 1 consider $(X_1,Y_1)=(U,U)$, for distribution 2 consider $(X_2,Y_2)=(U,V)$. Then define $f(x,x)=a$ and $f(x,y) = b$ whenever $x neq y$. So the difference is $|a-b|$ which is arbitrarily large.
$endgroup$
– Michael
Dec 27 '18 at 15:03
$begingroup$
Unfortunately the best bound is $(sup f) - (inf f)$. For example, for bounded $inf$ and $sup$, take $U, V$ iid uniform over $[0,1]$ and for distribution 1 consider $(X_1,Y_1)=(U,U)$, for distribution 2 consider $(X_2,Y_2)=(U,V)$. Then define $f(x,x)=a$ and $f(x,y) = b$ whenever $x neq y$. So the difference is $|a-b|$ which is arbitrarily large.
$endgroup$
– Michael
Dec 27 '18 at 15:03
add a comment |
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$begingroup$
Unfortunately the best bound is $(sup f) - (inf f)$. For example, for bounded $inf$ and $sup$, take $U, V$ iid uniform over $[0,1]$ and for distribution 1 consider $(X_1,Y_1)=(U,U)$, for distribution 2 consider $(X_2,Y_2)=(U,V)$. Then define $f(x,x)=a$ and $f(x,y) = b$ whenever $x neq y$. So the difference is $|a-b|$ which is arbitrarily large.
$endgroup$
– Michael
Dec 27 '18 at 15:03