This paper studies the minimum mean squared error (MMSE) of estimating X ∈ ℝd from the noisy observation Y ∈ ℝk, under the assumption that the noise (i.e., Y|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 Poincaré inequality to provide a new lower bound on the MMSE. Unlike, for example, the Cramér-Rao bound, the new bound holds for all possible distributions on the input X. Moreover, the lower bound is shown to be tight in the high-noise regime for the Gaussian noise setting under the assumption that X is sub-Gaussian. Finally, several numerical examples are shown which demonstrate that the bound performs well in all noise regimes.
|Original language||English (US)|
|Title of host publication||2022 IEEE International Symposium on Information Theory, ISIT 2022|
|Publisher||Institute of Electrical and Electronics Engineers Inc.|
|Number of pages||6|
|State||Published - 2022|
|Event||2022 IEEE International Symposium on Information Theory, ISIT 2022 - Espoo, Finland|
Duration: Jun 26 2022 → Jul 1 2022
|Name||2022 IEEE International Symposium on Information Theory (ISIT)|
|Conference||2022 IEEE International Symposium on Information Theory, ISIT 2022|
|Period||6/26/22 → 7/1/22|
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