[Paper Review] Optimal Robustness Results for Some Bayesian Procedures and the Relationship to Prior-Data Conflict

Luai Al‐Labadi, Michael Evans|arXiv (Cornell University)|Apr 27, 2015

Statistical Methods and Bayesian Inference14 references10 citations

TL;DR

This paper establishes optimal robustness properties for relative belief inferences in Bayesian statistics, showing they are robust when no prior-data conflict exists. It demonstrates that prior-data conflict undermines robustness, and provides quantitative measures to detect such conflict, reinforcing the need to assess it before inference.

ABSTRACT

The robustness to the prior of Bayesian inference procedures based on a measure of statistical evidence are considered. These inferences are shown to have optimal properties with respect to robustness. Furthermore, a connection between robustness and prior-data conflict is established. In particular, the inferences are shown to be effectively robust when the choice of prior does not lead to prior-data conflict. When there is prior-data conflict, however, robustness may fail to hold.

Motivation & Objective

• To establish optimal robustness of relative belief inferences under marginal prior variation over ε-contaminated priors.
• To resolve ambiguity in interpreting robustness results for relative belief ratios.
• To quantify sensitivity of inferences to both marginal and conditional priors.
• To clarify the connection between prior-data conflict and failure of robustness in Bayesian inference.
• To support the use of relative belief inferences as a preferred method for estimation and hypothesis assessment due to their robustness properties when prior-data conflict is absent.

Proposed method

• Uses relative belief ratios, defined as the ratio of posterior to prior density, to measure statistical evidence for a parameter value.
• Applies the Savage-Dickey ratio identity to express relative belief ratios in terms of prior and conditional predictive densities.
• Derives optimal robustness under ε-contaminated priors, generalizing prior results from Wasserman (1989) and others.
• Introduces quantitative sensitivity measures based on the ratio of predictive densities under different priors.
• Employs tail probability and ratio-based diagnostics to detect prior-data conflict in normal models.
• Uses simulation and numerical examples to illustrate how prior-data conflict leads to lack of robustness and extreme sensitivity in relative belief ratios.

Experimental results

Research questions

• RQ1 Under what conditions are relative belief inferences optimally robust to prior choice?
• RQ2 How is prior-data conflict related to the failure of robustness in Bayesian inference?
• RQ3 What quantitative measures can detect prior-data conflict and sensitivity of inferences to prior assumptions?
• RQ4 How does the behavior of the relative belief ratio change when prior-data conflict is present?
• RQ5 Can the relative belief ratio be meaningfully interpreted when prior-data conflict leads to large values?

Key findings

• Relative belief inferences are optimally robust to marginal prior variation over ε-contaminated priors, with robustness guaranteed when no prior-data conflict exists.
• In the presence of prior-data conflict—e.g., when the true parameter lies in the tails of the prior—robustness fails, as shown by extreme sensitivity in relative belief ratios.
• When data conflict with the prior on σ² but not μ, the ratio $ m_{Q,T}(T(x)|V(T(x)))/m_T(T(x)|V(T(x))) $ drops to 0.87 for μ₁ = -2, τ₁² = 1, indicating conflict and lack of robustness.
• In a case with conflict on μ but not σ², the ratio reaches 55,478,630 for μ₁ = 2, τ₁² = 1, signaling severe lack of robustness.
• The integral $ \int_{0}^{ u} RB((\bar{x},\sigma^{2})|x) \Pi_1(d\sigma^{-2}) = 8,046,933,962 $ in the conflict case indicates worst-case behavior and extreme sensitivity.
• The relative belief ratio at the maximum likelihood estimate diverges to infinity in continuous models, but large values in conflict scenarios signal lack of robustness, not high evidence.