A Statistical Assessment of Amortized Inference Under Signal-to-Noise Variation and Distribution Shift

Abstract: Since the turn of the century, approximate Bayesian inference has steadily evolved as new computational techniques have been incorporated to handle increasingly complex and large-scale predictive problems. The recent success of deep neural networks and foundation models has now given rise to a new paradigm in statistical modeling, in which Bayesian inference can be amortized through large-scale learned predictors. In amortized inference, substantial computation is invested upfront to train a neural network that can subsequently produce approximate posterior or predictions at negligible marginal cost across a wide range of tasks. At deployment, amortized inference offers substantial computational savings compared with traditional Bayesian procedures, which generally require repeated likelihood evaluations or Monte Carlo simulations for predictions for each new dataset. Despite the growing popularity of amortized inference, its statistical interpretation and its role within Bayesian inference remain poorly understood. This paper presents statistical perspectives on the working principles of several major neural architectures, including feedforward networks, Deep Sets, and Transformers, and examines how these architectures naturally support amortized Bayesian inference. We discuss how these models perform structured approximation and probabilistic reasoning in ways that yield controlled generalization error across a wide range of deployment scenarios, and how these properties can be harnessed for Bayesian computation. Through simulation studies, we evaluate the accuracy, robustness, and uncertainty quantification of amortized inference under varying signal-to-noise ratios and distributional shifts, highlighting both its strengths and its limitations.