In-Context Parametric Inference: Point or Distribution Estimators?

TL;DR

This paper compares Bayesian and frequentist inference methods in deep learning, revealing that amortized point estimators often outperform posterior inference across various problem settings, with insights into their trade-offs.

Contribution

It provides a rigorous comparative analysis of point and distribution estimators in amortized inference, highlighting their relative performance and generalization capabilities.

Findings

Amortized point estimators generally outperform posterior inference.

Posterior inference remains competitive in low-dimensional problems.

The study offers insights into the trade-offs between inference approaches.

Abstract

Bayesian and frequentist inference are two fundamental paradigms in statistical estimation. Bayesian methods treat hypotheses as random variables, incorporating priors and updating beliefs via Bayes' theorem, whereas frequentist methods assume fixed but unknown hypotheses, relying on estimators like maximum likelihood. While extensive research has compared these approaches, the frequentist paradigm of obtaining point estimates has become predominant in deep learning, as Bayesian inference is challenging due to the computational complexity and the approximation gap of posterior estimation methods. However, a good understanding of trade-offs between the two approaches is lacking in the regime of amortized estimators, where in-context learners are trained to estimate either point values via maximum likelihood or maximum a posteriori estimation, or full posteriors using normalizing flows, score-based diffusion samplers, or diagonal Gaussian approximations, conditioned on observations. To help resolve this, we conduct a rigorous comparative analysis spanning diverse problem settings, from linear models to shallow neural networks, with a robust evaluation framework assessing both in-distribution and out-of-distribution generalization on tractable tasks. Our experiments indicate that amortized point estimators generally outperform posterior inference, though the latter remain competitive in some low-dimensional problems, and we further discuss why this might be the case.