Abstract
Consider the problem of estimating a random variable X from noisy observations Y = X+ Z , where Z is standard normal, under the L1 fidelity criterion. It is well known that the optimal Bayesian estimator in this setting is the conditional median. This work shows that the only prior distribution on X that induces linearity in the conditional median is Gaussian. Along the way, several other results are presented. In particular, it is demonstrated that if the conditional distribution P_{X|Y=y} is symmetric for all y, then X must follow a Gaussian distribution. Additionally, we consider other Lp losses and observe the following phenomenon: for p in [1,2] , Gaussian is the only prior distribution that induces a linear optimal Bayesian estimator, and for p /in (2,∞) , infinitely many prior distributions on X can induce linearity. Finally, extensions are provided to encompass noise models leading to conditional distributions from certain exponential families.
Original language | English (US) |
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Pages (from-to) | 8026-8039 |
Number of pages | 14 |
Journal | IEEE Transactions on Information Theory |
Volume | 70 |
Issue number | 11 |
DOIs | |
State | Published - 2024 |
Externally published | Yes |
Keywords
- additive noise
- conditional mean
- Conditional median
- exponential family
- Fourier transform
- Gaussian noise
- input distribution
- mean absolute error
- mean square error
- poisson distribution
- posterior probability
- random variables
- tempered distributions
ASJC Scopus subject areas
- Information Systems
- Computer Science Applications
- Library and Information Sciences