[Paper Review] Combining probability distributions: Extending the logarithmic pooling approach

Luiz Max Carvalho, Daniel Antunes Maciel Villela+2 more|arXiv (Cornell University)|Feb 14, 2015

COVID-19 epidemiological studies40 references4 citations

TL;DR

This paper extends logarithmic pooling for combining probability distributions by introducing a hierarchical Bayesian approach to learn expert weights from data, enabling uncertainty quantification and prior-data conflict resolution. It demonstrates that hierarchical priors on weights improve robustness and adaptability in survival analysis, meta-analysis, and Bayesian melding of epidemic models.

ABSTRACT

Combining distributions is an important issue in decision theory and Bayesian inference. Logarithmic pooling is a popular method to aggregate expert opinions by using a set of weights that reflect the reliability of each information source. However, the resulting pooled distribution depends heavily on set of weights given to each opinion/prior and thus careful consideration must be given to the choice of weights. In this paper we review and extend the statistical theory of logarithmic pooling, focusing on the assignment of the weights using a hierarchical prior distribution. We explore several statistical applications, such as the estimation of survival probabilities, meta-analysis and Bayesian melding of deterministic models of population growth and epidemics. We show that it is possible learn the weights from data, although identifiability issues may arise for some configurations of priors and data. Furthermore, we show how the hierarchical approach leads to posterior distributions that are able to accommodate prior-data conflict in complex models.

Motivation & Objective

• Address the challenge of assigning reliable weights to expert opinions in probability distribution pooling, particularly when expert reliability is uncertain.
• Overcome limitations of fixed or optimality-based weight assignment by incorporating uncertainty in expert reliability through a hierarchical Bayesian model.
• Enable data-driven learning of pooling weights to improve consensus formation in complex models with potential prior-data conflict.
• Demonstrate the utility of hierarchical logarithmic pooling in practical statistical applications such as survival probability estimation and Bayesian melding.
• Provide theoretical and empirical justification for using hierarchical priors on weights to maintain coherence and flexibility in opinion pooling.

Proposed method

• Use logarithmic pooling to combine expert opinions via a weighted geometric mean of density functions: $\pi(\theta|\boldsymbol{\alpha}) = t(\boldsymbol{\alpha}) \prod_{i=0}^{K} f_i(\theta)^{\alpha_i}$, where $t(\boldsymbol{\alpha})$ ensures normalization.
• Introduce a hierarchical prior on the weight vector $\boldsymbol{\alpha}$, allowing the weights to be learned from data using Bayesian inference.
• Apply the hierarchical model to estimate weights in settings such as survival analysis, meta-analysis, and Bayesian melding of deterministic models.
• Leverage the property that log-concave component distributions yield log-concave pooled distributions, enabling efficient sampling via slice sampling or variational methods.
• Use posterior distributions over weights to detect and accommodate prior-data conflict, especially when expert priors contradict observed data.
• Compare hierarchical weighting to point-estimate methods (e.g., entropy maximization, KL divergence minimization), emphasizing the benefits of full posterior inference over weights.

Experimental results

Research questions

• RQ1 How can expert opinion pooling be made adaptive to data by learning the reliability weights rather than fixing them a priori?
• RQ2 What are the theoretical and practical implications of assigning hierarchical priors to the weights in logarithmic pooling?
• RQ3 In what ways does hierarchical weighting improve robustness to prior-data conflict in complex models like epidemic growth models?
• RQ4 How does the log-concavity of the pooled distribution affect computational efficiency and sampling in high-dimensional settings?
• RQ5 Can hierarchical logarithmic pooling outperform point-estimate methods such as entropy maximization or KL divergence minimization in terms of coherence and uncertainty quantification?

Key findings

• The hierarchical prior approach allows the weights to be learned from data, enabling sequential updating of expert reliability as new evidence emerges.
• The pooled distribution remains log-concave when all individual expert distributions are log-concave, preserving computational advantages for MCMC and variational inference.
• The method effectively resolves prior-data conflict by downweighting experts whose priors contradict the observed data, especially in Bayesian melding applications.
• The approach avoids violating Cromwell’s rule by assigning non-zero probability to all experts’ opinions, even when some are implausible.
• Empirical applications in survival analysis and meta-analysis show that hierarchical pooling leads to more coherent and robust posterior inferences than fixed-weight methods.
• The framework provides a principled alternative to point-estimate weight assignment, offering full posterior inference over weights and improved uncertainty quantification.