Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Generalized Power Priors for Improved Bayesian Inference with Historical Data

Masanari Kimura, Howard Bondell

TL;DR

The paper introduces a generalized power prior framework that uses alpha-divergence to robustly borrow historical data in Bayesian inference. The generalized power posterior is given by $g^*(\theta|D) \propto \left[ (1-\xi) p_0(\theta)^{(1+\alpha)/2} + \xi p_1(\theta)^{(1+\alpha)/2} \right]^{2/(1+\alpha)}$, showing it lies along the alpha-geodesic between the no-borrowing and full-borrowing pseudo-posteriors on the statistical manifold. The paper provides closed-form examples for Gaussian, Beta–Bernoulli, and Dirichlet–multinomial models, derives robustness bounds and consistency results, and offers an information-geometric interpretation. It demonstrates practical gains in survival analysis of ECOG melanoma trials using adaptive borrowing, balancing strength and robustness.

Abstract

The power prior is a class of informative priors designed to incorporate historical data alongside current data in a Bayesian framework. It includes a power parameter that controls the influence of historical data, providing flexibility and adaptability. A key property of the power prior is that the resulting posterior minimizes a linear combination of KL divergences between two pseudo-posterior distributions: one ignoring historical data and the other fully incorporating it. We extend this framework by identifying the posterior distribution as the minimizer of a linear combination of Amari's $α$-divergence, a generalization of KL divergence. We show that this generalization can lead to improved performance by allowing for the data to adapt to appropriate choices of the $α$ parameter. Theoretical properties of this generalized power posterior are established, including behavior as a generalized geodesic on the Riemannian manifold of probability distributions, offering novel insights into its geometric interpretation.

Generalized Power Priors for Improved Bayesian Inference with Historical Data

TL;DR

The paper introduces a generalized power prior framework that uses alpha-divergence to robustly borrow historical data in Bayesian inference. The generalized power posterior is given by , showing it lies along the alpha-geodesic between the no-borrowing and full-borrowing pseudo-posteriors on the statistical manifold. The paper provides closed-form examples for Gaussian, Beta–Bernoulli, and Dirichlet–multinomial models, derives robustness bounds and consistency results, and offers an information-geometric interpretation. It demonstrates practical gains in survival analysis of ECOG melanoma trials using adaptive borrowing, balancing strength and robustness.

Abstract

The power prior is a class of informative priors designed to incorporate historical data alongside current data in a Bayesian framework. It includes a power parameter that controls the influence of historical data, providing flexibility and adaptability. A key property of the power prior is that the resulting posterior minimizes a linear combination of KL divergences between two pseudo-posterior distributions: one ignoring historical data and the other fully incorporating it. We extend this framework by identifying the posterior distribution as the minimizer of a linear combination of Amari's -divergence, a generalization of KL divergence. We show that this generalization can lead to improved performance by allowing for the data to adapt to appropriate choices of the parameter. Theoretical properties of this generalized power posterior are established, including behavior as a generalized geodesic on the Riemannian manifold of probability distributions, offering novel insights into its geometric interpretation.

Paper Structure

This paper contains 19 sections, 11 theorems, 226 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Organization of the paper
  4. Backgroud
  5. Generalization
  6. Examples
  7. Theory
  8. Global Prior Sensitivity and Robustness against Outliers
  9. Shape of Generalized Power Posterior
  10. Consistency and Higher-order Asymptotic Analysis
  11. Information Geometry of Generalized Power Posterior
  12. Application
  13. Conclusion
  14. Auxiliary Results
  15. Proofs
  16. ...and 4 more sections

Key Result

Theorem 1

Let $\bm{\theta} \mapsto p_0(\bm{\theta})$ and $\bm{\theta} \mapsto p_1(\bm{\theta})$ be pseudo posteriors defined as follows. Here, the posterior $\bm{\theta} \mapsto g(\bm{\theta}) \coloneqq \pi(\bm{\theta} \mid D, D_0, \xi)$ is a minimizer of the following criterion. where $(p, q) \mapsto D_{\mathrm{KL}}[p \| q]$ is the KL divergence defined as

Figures (11)

  • Figure 1: Explicit examples of generalized power posterior. Univariate Gaussian case: $\bar{X} = 1$, $\bar{Y} = 2$, $\sigma^2=1$, $\mu_0=1$ and $\tau_0^2=1$. Beta-Bernoulli case: $\bar{X} = 0.6$, $\bar{Y}=0.7$, $\alpha_0 = 2$ and $\beta_0 = 8$. Dirichlet–Multinomial case: $\bm{\theta} = (20, 15, 15)^\top$ and $\bm{\alpha}_0 = (2.0, 2.0, 2.0)^\top$.
  • Figure 2: Upper bound $M_{\mathrm{TV}}(\alpha)$ of $d_{\mathrm{TV}}(g^*(\cdot; P_H), g^*(\cdot; P))$ with respect to $\alpha$.
  • Figure 3: Comparison of empirical $d_{\mathrm{TV}}$ and theoretical bounds.
  • Figure 4: Empirical $d_{TV}$ under reverse contamination.
  • Figure 5: Role of the parameter $\alpha$ for the shape of generalized power posterior (Gaussian distributions with same variance, $\xi = 0.5$). For the case of $\alpha \to \pm 1$, $\alpha = \pm 1 - 10^{-4}$ are used.
  • ...and 6 more figures

Theorems & Definitions (36)

  • Definition 1: Power prior ibrahim2000power
  • Theorem 1: Optimality of power prior under KL divergence Ibrahim01032003
  • Definition 2: Amari's $\alpha$-divergence amari2009alpha
  • Definition 3: Generalized power posterior
  • Proposition 1: Optimality of generalized power posterior under $\alpha$-divergence
  • Definition 4: Generalized power prior
  • Example 1: Univeriate Gaussian
  • Example 2: Beta–Bernoulli
  • Example 3: Dirichlet–Multinomial
  • Remark 1: Non-Conjugacy
  • ...and 26 more