Identifiability and amortized inference limitations in Kuramoto models

Since the turn of the century, approximate Bayesian inference has steadily evolved as new computational techniques have been incorporated to handle increasingly complex, 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 required at the beginning to train a neural network, but it can subsequently produce approximate posteriors or predictions at much lower computational cost across a wide range of tasks. While the typical Bayesian inference procedures are computationally expensive due to repeated likelihood calculations and Monte Carlo steps for each new dataset, amortized inference provides a much lower computational cost at deployment. Despite the growing popularity of amortized inference, its statistical interpretation and position within Bayesian inference remain poorly explored. In this paper, we present a statistical perspective on several major neural architectures, including feedforward networks, Deep Sets, and Transformers, and examine how they naturally support amortized Bayesian inference. We explore how these models perform structured approximation and also probabilistic reasoning in ways that yield controlled generalization error throughout a wide range of deployment scenarios, and how these properties can be harnessed for Bayesian computation. Via simulation studies, we evaluate the accuracy, robustness, and uncertainty quantification of amortized inference across varying sample sizes, varying noise distributional families, varying sparsity levels, and multimodality, highlighting its strengths and limitations.

Amortized inference promises fast test-time Bayesian inference, but existing methods typically target a fixed model or a restricted model family. Extending amortization to new model families often requires retraining or test-time fine-tuning. In this paper, we ask: is it possible to build a single inference network capable of generalizing across varying priors, likelihoods, latent dimensions, and observation counts? We introduce Amortized Factor Inference Networks (AFINs), which describe a model as a list of typed prior and likelihood factors and map this description, together with the observations, to a Gaussian posterior approximation through an encode--merge--decode network whose learned parameters do not depend on the latent dimension or the number of observations. Experiments on synthetic and real-data benchmarks show that a single trained AFIN provides competitive posterior approximations at substantially lower test-time cost than NUTS and per-task variational inference. Optional importance sampling can further improve accuracy.

We develop a general method for amortized Bayesian inference on multilevel models of arbitrary structure. Given a generative model specified as a directed acyclic graph, our method automatically derives valid factorizations of the joint posterior and matching neural network architectures. The key steps, graph expansion and graph inversion, yield an inverse graph that determines how inference networks are stacked and conditioned, producing factorizations that amortize over the number of groups and the number of observations within each group. Unlike approaches that simplify the dependency structure to speed up learning or inference, our method preserves all conditional independence and exchangeability assumptions of the generative model. Across three case studies, it closely matches gold-standard samplers on models with more than 6,500 parameters while reducing inference to a near-instant forward pass once trained.