Let $p,q$ be two distributions on $\mathbb{R}^d$. Let $f:\mathbb{R}^d\times\mathbb{R}^d\rightarrow\mathbb{R}$. Under what conditions does $\mathbb{E}_{x\sim p}f(x,z)=\mathbb{E}_{x\sim q}f(x,z)\ \forall z\in\mathbb{R}^d$ indicate $p=q$ a.e.? For example, can we prove $p=q$ a.e. when $f(x,z)=\tanh(x^{\top}z)$?

(Note: this problem may be related to discriminatory functions such as sigmoidal functions, but two distributions are involved in this case.)

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    $\begingroup$ So with that $f$ and with $d=1$, this is asking: if for all $z$, $$\int (p(x)-q(x))\tanh(xz)\, dx =0,$$ does it follow that $p=q$ as distributions? $\endgroup$ – Matt F. Oct 10 at 3:28
  • $\begingroup$ This is related to the moment problem. You need really strong conditions on $f$ and the space of possible distributions. $\endgroup$ – Christian Chapman Oct 10 at 3:29
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    $\begingroup$ @ChristianChapman, it is related but the moment problem is usually about only moments from integer powers, and the analog here would be asking whether distributions are determined by their moments from non-integer powers as well. Eg in the standard example (link.springer.com/content/pdf/10.1007%2F978-1-4419-5823-5_6.pdf) the lognormal and its perturbation only agree on moments for integral powers. $\endgroup$ – Matt F. Oct 10 at 3:35
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    $\begingroup$ The answer to Matt's version (his comment above) is clearly no since $\tanh xz$ is odd, so the integral will be zero for all $z$ as soon as $p-q$ is even (so in particular, for any two even $p,q$). $\endgroup$ – Christian Remling Oct 10 at 14:45
  • $\begingroup$ I found a useful material. See Lemma 1 in jmlr.org/papers/volume13/gretton12a/gretton12a.pdf $\endgroup$ – Zhifeng Kong Oct 11 at 4:10

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