Let $X$ be a real random variable, and, for each $t \geq 0$, let $X_t = X + \sqrt {t}Y$ where $Y$ is a standard normal independent of $X$. The quantity $$ R(t) = E \big ( [ X - E (X |X_t)]^2 \big), \quad t \geq 0, $$ which describes the quadratic discrepancy of a random variable from its observation perturbated by a Gaussian noise, is known as the minimum mean-square error in Gaussian channel in information theory. It seems natural to conjecture that the function $R(t)$, $t \geq 0$, characterizes the distribution of $X$, up to translation and symmetry (clearly $R$ is invariant if $X$ is replaced by $X + c$ or by $-X$). Is there anything known in this regard?
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$\begingroup$ Tweedie's formula gives explicit expressions for the conditional mean $E[X|X_t]$ and variance $Var[X|X_t]$. These can be expressed in terms of the first and second derivatives of $\log f$ takien at $X_t$ where $f$ the density of $X$. If you know $R(t)=E[Var[X|X_t]]$, taking derivatives a $0^+$ (and maybe the Taylor expansion of $R(t)$ at 0) should provide information about certain moments of the derivatives of $f$. I don't know if that will be sufficient to characterize the distribution of $X$. $\endgroup$– jlewkCommented Oct 20, 2021 at 1:40
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