Estimation Error: Distribution and Pointwise Limits
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915380224589824 |
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| author | Barletta, Luca Dytso, Alex Shamai, Shlomo |
| author_facet | Barletta, Luca Dytso, Alex Shamai, Shlomo |
| contents | In this paper, we examine the distribution and convergence properties of the estimation error $W = X - \hat{X}(Y)$, where $\hat{X}(Y)$ is the Bayesian estimator of a random variable $X$ from a noisy observation $Y = X +σZ$ where $σ$ is the parameter indicating the strength of noise $Z$. Using the conditional expectation framework (that is, $\hat{X}(Y)$ is the conditional mean), we define the normalized error $\mathcal{E}_σ= \frac{W}σ$ and explore its properties.
Specifically, in the first part of the paper, we characterize the probability density function of $W$ and $\mathcal{E}_σ$. Along the way, we also find conditions for the existence of the inverse functions for the conditional expectations. In the second part, we study pointwise (i.e., almost sure) convergence of $\mathcal{E}_σ$ as $σ\to 0$ under various assumptions about the noise and the underlying distributions. Our results extend some of the previous limits of $\mathcal{E}_σ$ as $σ\to 0$ studied under the $L^2$ convergence, known as the \emph{mmse dimension}, to the pointwise case. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_11109 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Estimation Error: Distribution and Pointwise Limits Barletta, Luca Dytso, Alex Shamai, Shlomo Information Theory In this paper, we examine the distribution and convergence properties of the estimation error $W = X - \hat{X}(Y)$, where $\hat{X}(Y)$ is the Bayesian estimator of a random variable $X$ from a noisy observation $Y = X +σZ$ where $σ$ is the parameter indicating the strength of noise $Z$. Using the conditional expectation framework (that is, $\hat{X}(Y)$ is the conditional mean), we define the normalized error $\mathcal{E}_σ= \frac{W}σ$ and explore its properties. Specifically, in the first part of the paper, we characterize the probability density function of $W$ and $\mathcal{E}_σ$. Along the way, we also find conditions for the existence of the inverse functions for the conditional expectations. In the second part, we study pointwise (i.e., almost sure) convergence of $\mathcal{E}_σ$ as $σ\to 0$ under various assumptions about the noise and the underlying distributions. Our results extend some of the previous limits of $\mathcal{E}_σ$ as $σ\to 0$ studied under the $L^2$ convergence, known as the \emph{mmse dimension}, to the pointwise case. |
| title | Estimation Error: Distribution and Pointwise Limits |
| topic | Information Theory |
| url | https://arxiv.org/abs/2501.11109 |