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New error bounds for M-testing and estimation of source location with subdiffractive error.

Sudhakar Prasad1

  • 1Department of Physics and Astronomy, University of New Mexico, Albuquerque, New Mexico 87131, USA. sprasad@unm.edu

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|April 5, 2012
PubMed
Summary
This summary is machine-generated.

New bounds on the minimum probability of error (MPE) in Bayesian multihypothesis testing are derived. These bounds are exponentially tight for many measurements and relate MPE to minimum mean-squared error (MMSE).

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Area of Science:

  • Information Theory
  • Statistical Inference
  • Bayesian Analysis

Background:

  • Bayesian multihypothesis testing is crucial for statistical decision-making.
  • Quantifying the minimum probability of error (MPE) is a fundamental challenge.
  • Understanding the relationship between MPE and minimum mean-squared error (MMSE) is important.

Purpose of the Study:

  • To establish new lower and upper bounds on the MPE in Bayesian multihypothesis testing.
  • To demonstrate the asymptotic achievability and exponential tightness of these bounds.
  • To explore the connection between MPE and MMSE, particularly in point source localization.

Main Methods:

  • Exact integration of statistical entropy (equivocation) of the posterior distribution.
  • Asymptotic analysis for a large number of conditionally independent and identically distributed measurements.
  • Derivation of error probability integrals relating MPE and MMSE.

Main Results:

  • New exponentially tight lower and upper bounds on MPE were derived.
  • These bounds are achievable in the asymptotic limit.
  • A relationship between MPE and MMSE was established using error probability integrals.
  • Comparison of MPE and MMSE for subdiffractive point source localization revealed a more modest source-strength threshold than for optical super-resolution.

Conclusions:

  • The derived bounds provide a powerful tool for analyzing error probabilities in Bayesian hypothesis testing.
  • The findings offer insights into the fundamental limits of information detection and estimation.
  • The study highlights potential advantages in source localization using MPE compared to MMSE under certain conditions.