[Paper Review] Uncertainty quantification using martingales for misspecified Gaussian processes

Willie Neiswanger, Aaditya Ramdas|arXiv (Cornell University)|Jun 12, 2020

Gaussian Processes and Bayesian Inference37 references5 citations

TL;DR

This paper proposes a frequentist confidence sequence for Gaussian processes (GPs) that remains statistically valid under prior misspecification, using martingale techniques to construct uniformly valid uncertainty bands. Unlike standard GP posterior bands, which can be misleading when the prior is incorrect, the proposed method ensures coverage probability guarantees regardless of prior accuracy, at the cost of wider intervals when the prior is correct—demonstrating robustness in practice, especially in Bayesian optimization settings.

ABSTRACT

We address uncertainty quantification for Gaussian processes (GPs) under misspecified priors, with an eye towards Bayesian Optimization (BO). GPs are widely used in BO because they easily enable exploration based on posterior uncertainty bands. However, this convenience comes at the cost of robustness: a typical function encountered in practice is unlikely to have been drawn from the data scientist's prior, in which case uncertainty estimates can be misleading, and the resulting exploration can be suboptimal. We present a frequentist approach to GP/BO uncertainty quantification. We utilize the GP framework as a working model, but do not assume correctness of the prior. We instead construct a confidence sequence (CS) for the unknown function using martingale techniques. There is a necessary cost to achieving robustness: if the prior was correct, posterior GP bands are narrower than our CS. Nevertheless, when the prior is wrong, our CS is statistically valid and empirically outperforms standard GP methods, in terms of both coverage and utility for BO. Additionally, we demonstrate that powered likelihoods provide robustness against model misspecification.

Motivation & Objective

• To address the problem of unreliable uncertainty estimates in Gaussian processes when the prior is misspecified, a common issue in Bayesian optimization.
• To develop a frequentist approach that provides uniformly valid confidence sequences for the true function, independent of prior correctness.
• To maintain statistical validity in high-dimensional or infinite-dimensional function spaces where the Bernstein-von Mises theorem fails.
• To evaluate the trade-off between robustness and precision in uncertainty quantification, particularly in scenarios where the prior is inaccurate.
• To explore practical integration of hyperparameter tuning and adaptive belief parameters within the confidence sequence framework.

Proposed method

• Construct a confidence sequence (CS) for the unknown function using martingale theory, ensuring that the true function lies within the sequence with high probability uniformly over time.
• Use a function class $\mathcal{F}$ and a belief parameter $\gamma$ to define a stochastic process that tracks the discrepancy between observed data and predictions under the prior.
• Apply sequential probability ratio tests and martingale inequalities to derive data-dependent confidence bands that adaptively widen or narrow based on evidence.
• Integrate powered likelihoods to enhance robustness against model misspecification, particularly when hyperparameters are estimated from data.
• Implement the method using a kernelized function class and compute the confidence set $C_t$ at each time step $t$ via a convex combination of kernel-based projections.
• Ensure computational feasibility by selecting $\mathcal{F}$ such that the confidence set remains tractable, even under misspecification.

Experimental results

Research questions

• RQ1 Can a confidence sequence be constructed for Gaussian processes that maintains valid coverage under arbitrary prior misspecification?
• RQ2 How does the performance of the proposed confidence sequence compare to standard GP posterior bands in terms of coverage and utility for Bayesian optimization?
• RQ3 What is the trade-off between robustness and precision when using confidence sequences instead of posterior credible intervals?
• RQ4 Can hyperparameter estimation methods like maximum likelihood or cross-validation be meaningfully integrated into the confidence sequence framework?
• RQ5 How can the belief parameter $\gamma$ be adapted dynamically to balance trust in the prior with robustness to misspecification?

Key findings

• The proposed confidence sequence maintains uniform coverage probability $\leq \alpha$ over time, even when the GP prior is misspecified, ensuring statistical validity.
• When the prior is correct, the confidence sequence is wider than standard GP posterior bands, reflecting the cost of robustness.
• Under prior misspecification, the confidence sequence recovers and contains the true function as more data is collected, while standard GP bands fail to converge to the truth.
• The method outperforms standard GP uncertainty quantification in terms of both coverage and utility in Bayesian optimization simulations.
• Powered likelihoods are shown to improve robustness against model misspecification, particularly when hyperparameters are estimated from data.
• The results confirm that the Bernstein-von Mises theorem does not hold in infinite-dimensional function spaces, justifying the need for frequentist confidence sequences in such settings.