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Bayesian meta learning for trustworthy uncertainty quantification

Zhenyuan Yuan, Thinh T. Doan

TL;DR

The paper addresses trustworthy uncertainty quantification in Bayesian regression by defining ground-truth capture within predictive intervals and introducing Trust-Bayes, a Bayesian meta-learning framework that enforces these constraints using meta-training data without relying on explicit prior models of the target functions. It develops finite-sample, empirical-bounds-based guarantees on the inclusion probabilities for both prior and posterior intervals through a theorem that leverages 0-1 losses, sub-Gaussian concentration, and evaluation data. The method parameterizes priors as $m_\theta$ and $k_\phi$, with a scalable kernel, and optimizes an objective like negative marginal log-likelihood subject to the trustworthy-quantification constraints. A Gaussian process regression case study demonstrates that Trust-Bayes achieves reliable coverage and outperforms Meta-prior when priors are mis-specified, highlighting its practical impact for safe, data-efficient learning in uncertainty-sensitive applications.

Abstract

We consider the problem of Bayesian regression with trustworthy uncertainty quantification. We define that the uncertainty quantification is trustworthy if the ground truth can be captured by intervals dependent on the predictive distributions with a pre-specified probability. Furthermore, we propose, Trust-Bayes, a novel optimization framework for Bayesian meta learning which is cognizant of trustworthy uncertainty quantification without explicit assumptions on the prior model/distribution of the functions. We characterize the lower bounds of the probabilities of the ground truth being captured by the specified intervals and analyze the sample complexity with respect to the feasible probability for trustworthy uncertainty quantification. Monte Carlo simulation of a case study using Gaussian process regression is conducted for verification and comparison with the Meta-prior algorithm.

Bayesian meta learning for trustworthy uncertainty quantification

TL;DR

The paper addresses trustworthy uncertainty quantification in Bayesian regression by defining ground-truth capture within predictive intervals and introducing Trust-Bayes, a Bayesian meta-learning framework that enforces these constraints using meta-training data without relying on explicit prior models of the target functions. It develops finite-sample, empirical-bounds-based guarantees on the inclusion probabilities for both prior and posterior intervals through a theorem that leverages 0-1 losses, sub-Gaussian concentration, and evaluation data. The method parameterizes priors as and , with a scalable kernel, and optimizes an objective like negative marginal log-likelihood subject to the trustworthy-quantification constraints. A Gaussian process regression case study demonstrates that Trust-Bayes achieves reliable coverage and outperforms Meta-prior when priors are mis-specified, highlighting its practical impact for safe, data-efficient learning in uncertainty-sensitive applications.

Abstract

We consider the problem of Bayesian regression with trustworthy uncertainty quantification. We define that the uncertainty quantification is trustworthy if the ground truth can be captured by intervals dependent on the predictive distributions with a pre-specified probability. Furthermore, we propose, Trust-Bayes, a novel optimization framework for Bayesian meta learning which is cognizant of trustworthy uncertainty quantification without explicit assumptions on the prior model/distribution of the functions. We characterize the lower bounds of the probabilities of the ground truth being captured by the specified intervals and analyze the sample complexity with respect to the feasible probability for trustworthy uncertainty quantification. Monte Carlo simulation of a case study using Gaussian process regression is conducted for verification and comparison with the Meta-prior algorithm.
Paper Structure (6 sections, 5 theorems, 20 equations, 3 figures, 1 table)

This paper contains 6 sections, 5 theorems, 20 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. The Trust-Bayes framework
  4. Proof of Theorem \ref{['thm: PP']}
  5. Case study
  6. Conclusion

Key Result

Theorem III.1

Suppose $\{f^i\}_{i=1}^n$ are sampled i.i.d. from a latent distribution $\mathcal{P}_f$. Suppose the training dataset $\mathcal{D}^i_{tr}$ for each function $f^i$ is generated through a latent conditional distribution $\mathcal{P}_{\mathcal{D}^i_{tr}\mid f^i}$, i.e., $\mathcal{D}^i_{tr}\sim\mathcal{ $\blacksquare$

Figures (3)

  • Figure 1: Comparison over empirical estimates of $P_{f^i,x}(f^i(x)\in\mathcal{I}_{\theta,\phi}(x))$ and $P_{f^i,x}(f^i(x)\in\mathcal{I}^i_{\theta,\phi}(x))$ by the evaluation dataset $\{\mathcal{D}^{i}_{eval}\}_{i=1}^n$ between Trust-Bayes and Meta-prior during meta-training
  • Figure 2: Ten random functions drawn from \ref{['eq: simulation functions']} and comparison between prior predictions by Trust-Bayes and Meta-prior
  • Figure 3: Comparison between posterior predictions by Trust-Bayes and Meta-prior over ten random functions from \ref{['eq: simulation functions']}

Theorems & Definitions (8)

  • Remark II.1
  • Theorem III.1
  • Remark III.1
  • Definition IV.1
  • Lemma IV.2
  • Lemma IV.3
  • Lemma IV.4
  • Lemma IV.5