An MMSE Lower Bound via Poincar\'e Inequality. (arXiv:2205.05848v1 [cs.IT])

This paper studies the minimum mean squared error (MMSE) of estimating
$\mathbf{X} \in \mathbb{R}^d$ from the noisy observation $\mathbf{Y} \in
\mathbb{R}^k$, under the assumption that the noise (i.e.,
$\mathbf{Y}|\mathbf{X}$) is a member of the exponential family. The paper
provides a new lower bound on the MMSE. Towards this end, an alternative
representation of the MMSE is first presented, which is argued to be useful in
deriving closed-form expressions for the MMSE. This new representation is then
used together with the …

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