Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Testing the Isotropic Cauchy Hypothesis

Jihad Fahs, Ibrahim Abou-Faycal, Ibrahim Issa

TL;DR

The paper investigates binary hypothesis testing between isotropic Gaussian and isotropic Cauchy in high dimensions, where the Cauchy model induces correlation and yields an infinite relative entropy D(p_C||p_G). The authors show that both Bayesian and Neyman-Pearson detectors exhibit non-exponential error decay, with the Bayesian error scaling as P_e = Θ(√(ln n / n)) and NP analysis revealing a 1/√n decay for P_G|C when P_F is fixed, along with exponential bounds when the miss probability is fixed. The analysis hinges on a detailed operating-regime decomposition driven by the likelihood-ratio geometry and Lambert W functions, revealing that correlation fundamentally alters classical IID exponents. These insights suggest broader implications for hypothesis testing when p_n and q_n are not IID and where D(p_n||q_n) grows only logarithmically with n. The results are validated numerically and point to further exploration of D(p_n||q_n) in dependent settings and potential extensions to other heavy-tailed isotropic models.

Abstract

Isotropic $α$-stable distributions are central in the theory of heavy-tailed distributions and play a role similar to that of the Gaussian density among finite second-moment laws. Given a sequence of $n$ observations, we are interested in characterizing the performance of Likelihood Ratio Tests where two hypotheses are plausible for the observed quantities: either isotropic Cauchy or isotropic Gaussian. Under various setups, we show that the probability of error of such detectors is not always exponentially decaying with $n$ with the leading term in the exponent shown to be logarithmic instead and we determine the constants in that leading term. Perhaps surprisingly, the optimal Bayesian probabilities of error are found to exhibit different asymptotic behaviors.

Testing the Isotropic Cauchy Hypothesis

TL;DR

The paper investigates binary hypothesis testing between isotropic Gaussian and isotropic Cauchy in high dimensions, where the Cauchy model induces correlation and yields an infinite relative entropy D(p_C||p_G). The authors show that both Bayesian and Neyman-Pearson detectors exhibit non-exponential error decay, with the Bayesian error scaling as P_e = Θ(√(ln n / n)) and NP analysis revealing a 1/√n decay for P_G|C when P_F is fixed, along with exponential bounds when the miss probability is fixed. The analysis hinges on a detailed operating-regime decomposition driven by the likelihood-ratio geometry and Lambert W functions, revealing that correlation fundamentally alters classical IID exponents. These insights suggest broader implications for hypothesis testing when p_n and q_n are not IID and where D(p_n||q_n) grows only logarithmically with n. The results are validated numerically and point to further exploration of D(p_n||q_n) in dependent settings and potential extensions to other heavy-tailed isotropic models.

Abstract

Isotropic -stable distributions are central in the theory of heavy-tailed distributions and play a role similar to that of the Gaussian density among finite second-moment laws. Given a sequence of observations, we are interested in characterizing the performance of Likelihood Ratio Tests where two hypotheses are plausible for the observed quantities: either isotropic Cauchy or isotropic Gaussian. Under various setups, we show that the probability of error of such detectors is not always exponentially decaying with with the leading term in the exponent shown to be logarithmic instead and we determine the constants in that leading term. Perhaps surprisingly, the optimal Bayesian probabilities of error are found to exhibit different asymptotic behaviors.
Paper Structure (16 sections, 6 theorems, 63 equations, 4 figures)

This paper contains 16 sections, 6 theorems, 63 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Main Results
  4. Numerical Simulations
  5. Technical Proofs
  6. Preliminaries
  7. Operating Regime (OR)
  8. The Bayesian Detection Problem; Proof
  9. The Neyman-Pearson Formulation; Proof
  10. Conclusion
  11. The KL Divergence
  12. Correlated Hypotheses; Examples
  13. Example 1
  14. Example 2: The Beta-Binomial Distribution
  15. Example 3: Continuous Laws
  16. ...and 1 more sections

Key Result

Theorem 1

Consider a sequence of $n$ observations and a binary decision problem with two hypotheses for their distribution: either circular (IID) Gaussian $\mathcal{N}(0,\sigma^2)$ with prior probability $0 < \pi_G < 1$, or circular (correlated) Cauchy $\mathcal{C}(0,\gamma)$ with prior probability $\pi_C = 1

Figures (4)

  • Figure 1: $P_e$ vs. $n$ for $\eta = 0.5, 1, 1.5$. Solid line: numerical computations. Dashed line: approximate expression.
  • Figure 2: $(P_{\text{G}|\text{C}})$ vs. n for $P_F = 0.01, 0.05, 0.09$. Solid line: numerical computations. Dashed line: approximate expression.
  • Figure 3: $- \ln (P_{\text{C}|\text{G}})$ vs. n for $1- P_D = 0.07, 0.14$.
  • Figure 4: $\ell(r)$ for $\xi^2=1$, with $n=5$ (solid blue), $n=8$ (dashed green), and $n=13$ (pointed red).

Theorems & Definitions (10)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • Lemma 2
  • proof
  • Theorem 2
  • proof