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Weighted Chernoff information and optimal loss exponent in context-sensitive hypothesis testing

Mark Kelbert, El'mira Yu. Kalimulina

TL;DR

This work considers context-sensitive (binary) hypothesis testing for i.i.d. observations under a multiplicative weight function and embeds weighted geometric mixtures into an exponential family and identifies the exponent as the maximizer of its log-normaliser.

Abstract

We consider context-sensitive (binary) hypothesis testing for i.i.d. observations under a multiplicative weight function. We establish the logarithmic asymptotic, as the sample size grows, of the optimal total loss (sum of type-I and type-II losses) and express the corresponding error exponent through a weighted Chernoff information between the competing distributions. Our approach embeds weighted geometric mixtures into an exponential family and identifies the exponent as the maximizer of its log-normaliser. We also provide concentration bounds for a tilted weighted log-likelihood and derive explicit expressions for Gaussian and Poisson models, as well as further parametric examples.

Weighted Chernoff information and optimal loss exponent in context-sensitive hypothesis testing

TL;DR

This work considers context-sensitive (binary) hypothesis testing for i.i.d. observations under a multiplicative weight function and embeds weighted geometric mixtures into an exponential family and identifies the exponent as the maximizer of its log-normaliser.

Abstract

We consider context-sensitive (binary) hypothesis testing for i.i.d. observations under a multiplicative weight function. We establish the logarithmic asymptotic, as the sample size grows, of the optimal total loss (sum of type-I and type-II losses) and express the corresponding error exponent through a weighted Chernoff information between the competing distributions. Our approach embeds weighted geometric mixtures into an exponential family and identifies the exponent as the maximizer of its log-normaliser. We also provide concentration bounds for a tilted weighted log-likelihood and derive explicit expressions for Gaussian and Poisson models, as well as further parametric examples.
Paper Structure (24 sections, 19 theorems, 141 equations)

This paper contains 24 sections, 19 theorems, 141 equations.

Table of Contents

  1. Introduction
  2. Main result and contributions.
  3. Problem set-up and weighted divergences
  4. Context-sensitive losses and weighted total variation
  5. Weighted affinities and Chernoff information
  6. Main results
  7. Asymptotics of the optimal sum of losses
  8. Exponential-family representation and uniqueness of $\alpha^*$
  9. Weighted Bregman divergence and information-geometric identities
  10. Weighted KL divergence and weighted Bregman divergence.
  11. Weighted Chernoff/Bhattacharyya quantities inside an exponential family.
  12. Derivative and weighted KL.
  13. Chernoff--KL.
  14. Tilted weighted likelihood and concentration bounds
  15. Non-asymptotic concentration via a Doob martingale.
  16. ...and 9 more sections

Key Result

Proposition 2.1

For each $n\ge 1$, Moreover, an optimal (deterministic) decision rule is given by the likelihood-ratio test (with any measurable tie-breaking on $\{p=q\}$).

Theorems & Definitions (51)

  • Proposition 2.1: Pointwise form of the optimal total loss
  • proof
  • Remark 2.2
  • Definition 2.3: Weighted Bhattacharyya coefficient and distance
  • Definition 2.4: Weighted Chernoff information
  • Remark 2.5
  • Remark 2.6
  • Theorem 3.1: Optimal sum of context-sensitive losses
  • proof
  • Corollary 3.2: Asymptotics of the weighted total variation
  • ...and 41 more