Neural posterior estimation has emerged as a powerful tool for amortized inference, with growing adoption across scientific and applied domains. In many of these applications, the conditioning variable is a set of observations whose elements depend not only on the target but also on unknown factors shared across the set. Optimal inference therefore requires treating the set jointly, which in turn requires training the estimator at the deployment set size -- a regime where memory and compute quickly become prohibitive. We introduce a simple, theoretically grounded strategy that decouples representation learning from posterior modeling. Our method trains a mean-pool Deep Set on sets of size at most two, producing an encoder that generalizes to arbitrary set sizes. The inference head is then finetuned on pre-aggregated embeddings, making training cost essentially independent of the deployment set size N. Across scalar, image, multi-view 3D, molecular, and high-dimensional conditional generation benchmarks with N in the thousands, our approach matches or outperforms standard baselines at a fraction of the compute.
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.
Roy Shivam Ram Shreshtth, Arnab Hazra, Gourab Mukherjee
Amortized inference promises fast test-time Bayesian inference, but existing methods are inherently tied to fixed models. Extending amortization to unseen models typically requires retraining or costly test-time finetuning. In this paper, we ask: is it possible to build a single inference network capable of generalizing across varying priors, likelihoods, and dimensionality? We introduce Amortized Factor Inference Networks (AFINs), a family of encode-merge-decode inference networks built on dimension-independent modules that map a model specification and its observations to the parameters of a variational posterior. Experimentally, a single trained AFIN achieves posterior accuracy comparable to NUTS and several variational inference methods, while requiring 2 to 4 orders of magnitude less test-time compute. Code is available at https://github.com/joohwanko/AFINs.
Learning neural set functions is pivotal to a wide range of important applications, including compound selection in AI-driven drug discovery and product recommendation. Recent work has introduced optimal subset oracles to implicitly learn set functions under practical weakly supervised settings, where model parameters are optimized through mean-field variational inference. However, these frameworks rely on Monte Carlo sampling to estimate gradients of the evidence lower bound when updating the variational distribution. Repeated sampling across iterations incurs substantial computational overhead, while the resulting stochasticity can destabilize the optimization trajectory. In this work, we reinterpret the evidence lower bound as a continuous relaxation of the set function and learn a surrogate objective that replaces sampling-based ELBO gradient estimation during variational optimization. The learned surrogate provides stable and efficient gradients throughout the continuous domain, thereby reducing computational overhead and accelerating inference. Furthermore, we establish an approximation guarantee for the proposed framework under submodular maximization and characterize its connection to variational free energy. Experiments on a variety of real-world tasks demonstrate consistent improvements over existing baselines.