Generative Modeling: A Review

We organize the generative-modeling literature around three classes of generators, corresponding to three distinct inferential tasks: estimating counterfactual outcome distributions in causal inference, recovering posteriors from simulated parameter--outcome pairs, and forming predictive outcome distributions. The unifying representation relies on the noise outsourcing theorem of Kallenberg, which expresses a conditional distribution as a deterministic function of its inputs and an independent noise variable. Within this organization we develop generative Bayesian computation, a method in the parameter--outcome class: a quantile neural network, trained on simulated pairs under the pinball loss, that targets the posterior of the parameter directly, without invertible architectures or density evaluation, and that serves equally as a predictive generator once the roles of parameter and outcome are exchanged. We illustrate the framework on an agent-based Ebola transmission application, where generative Bayesian computation recovers accurate posteriors at substantially lower cost than rejection-based simulation inference, while avoiding the density-evaluation and invertibility constraints of competing generators.