Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A universal robust bound for the intrinsic Bayes factor

Richard Clare

Abstract

In this work, we undertake a comprehensive reformulation, modification, and extension of Smith and Spiegelhalter's (1980) and (1982) Bayes Factor work within the evolving subject of Objective Bayes Factors. Our primary focus centers on defining and computing empirical and theoretical bounds for the Intrinsic Bayes Factor (IBF) across various models, including normal, exponential, Poisson, geometric, linear, and ANOVA. We show that our new bounds are useful, feasible, and change with the amount of information. We also propose a methodology to construct the least favorable (for the null model) intrinsic priors that result in the lower and upper bounds of the Intrinsic Bayes Factors under certain conditions. Notably, our lower bounds exhibit superior performance compared to the well-known -ep log(p) bound proposed by Sellke et al. (2001) (Sellke et al., 2001) based on p-values.

A universal robust bound for the intrinsic Bayes factor

Abstract

In this work, we undertake a comprehensive reformulation, modification, and extension of Smith and Spiegelhalter's (1980) and (1982) Bayes Factor work within the evolving subject of Objective Bayes Factors. Our primary focus centers on defining and computing empirical and theoretical bounds for the Intrinsic Bayes Factor (IBF) across various models, including normal, exponential, Poisson, geometric, linear, and ANOVA. We show that our new bounds are useful, feasible, and change with the amount of information. We also propose a methodology to construct the least favorable (for the null model) intrinsic priors that result in the lower and upper bounds of the Intrinsic Bayes Factors under certain conditions. Notably, our lower bounds exhibit superior performance compared to the well-known -ep log(p) bound proposed by Sellke et al. (2001) (Sellke et al., 2001) based on p-values.
Paper Structure (52 sections, 4 theorems, 238 equations, 12 figures, 1 algorithm)

This paper contains 52 sections, 4 theorems, 238 equations, 12 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. LITERATURE REVIEW
  3. Bayes factors
  4. Prior and Posterior Model Probabilities
  5. Jeffreys's objective priors
  6. Objective Bayes Factors
  7. Minimal training samples
  8. A formal definition for the set of all minimal training samples
  9. An algorithm to generate the set of all minimal training samples
  10. The Intrinsic Bayes Factor
  11. The Expected Posterior Bayes Factors
  12. A UNIVERSAL ROUBUST BOUND
  13. The Smith and Spiegelhalter's method
  14. The universal robust bound for the Intrinsic Bayes Factors
  15. A bridge between the universal bound and the IBF
  16. ...and 37 more sections

Key Result

Lemma 2.6.1

Let $\textbf{y} = \{y_1,...,y_n\}$ be independent and identically distributed random samples, and assume that $\textbf{y}(\ell) \in D$ with $\textbf{y}(-\ell)\cup \textbf{y}(\ell) = \textbf{y}$, then

Figures (12)

  • Figure 1: A comparison between the SP Prior and the Intrinsic Prior for the Normal Scale Hypothesis test for different null hypothesis values.
  • Figure 2: $\overline{B}^{SS}_{10}$ vs $B^{SP}_{10}$ under $H_0$ for increasing sample size and exponential data.
  • Figure 3: Simulations with different number of samples for the exponential SP and EP Priors for $\lambda_0=1$
  • Figure 4: Logarithm of the SP and EP Bayes factor using one-hundred exponential simulations (100 samples each) for $\lambda_0 = 1, H_0$ True
  • Figure 5: Logarithm of the SP and EP Bayes factor using one-hundred exponential simulations (100 samples each) for $\lambda_0 = 1, H_0$ False
  • ...and 7 more figures

Theorems & Definitions (15)

  • Lemma 2.6.1
  • proof
  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 3.3.1
  • proof
  • Definition 4: The Theoretical SP Prior
  • Definition 5: The Empirical SP Prior
  • Definition 6: The Theoretical SP Bayes Factor
  • ...and 5 more