ForwardFlow: Simulation only statistical inference using deep learning

We present a new perspective on the problem of likelihood approximation within the framework of simulation-based inference that promotes scalable and controllable simulation routines for large-scale data analysis, allows efficient parameter space exploration or smooth interpolation in high-dimensions and, thus, supports valid statistical treatments of hypothesis testings as well as uncertainty quantification. In particular, we consider a chain of $n$-aggregated normalizing flows for likelihood approximation scheme, where a set of upfront replicated observation datasets from the forward complex simulation model pass through the first set of bijective transformations, and then subsequently pass to the other sets of bijective transformations. Here, we assume that, for any $k \in \{1,\,2, \ldots, n\}$, the parameters corresponding to the first $k$ sets of bijective transformations are estimated sequentially, in some sense of optimality, for constructing flexible probability distributions, regardless of the remaining $(n-k)$ sets of bijective transformations. Moreover, our objects of interest are to highlight two complementary mathematical arguments that leverage an informatics-theoretic formalization, based-on empirical likelihood estimators under moment restrictions, and a sequential decision-making paradigm, with mixing distributions, for updating and aggregating the estimated parameters of the overall normalizing flows. As a by-product, the framework provides a reliable surrogate model, conditioned on the model parameters defining the forward computational simulation, that allows samples generation, with statistical powers, and facilitates computationally tractable scheme in the Bayesian paradigm for inference, hypothesis testings and uncertainty quantification.

In this work, we present a simulation-based parameter estimation framework for a model defined by a computational simulation of a physical system. We specifically outline an estimation framework consisting of two closely-integrated steps that facilitate an overall end-to-end parameter estimation scheme. The first step involves utilizing an embedded normalizing flow which is used to transform the unknown complex distribution of the residual information into a simple base distribution corresponding to the transformed residual information. In the second step, an empirical-likelihood estimator, under moment restrictions, is utilized for imposing an indirect constrain on the base distribution, where such an instantiated task reasonably allows us to treat the transformed residual information as random variables arising from discretely distribution population with each transformed data point as a single-cell from a set of finite-cell contingencies. Moreover, we use first-order gradient methods for updating the estimated parameter values of the model defined by the computational simulation and the corresponding parametrized embedded normalizing flow, that call for all gradient-related information by leveraging implicitly differentiations of the empirical-likelihood function, which is constructed from the implied empirical probabilities under moment restrictions. Here, it is worth mentioning that the problem formulation presented in this work, which highlights an information-theoretic interpretation, allows to present a computational framework for algorithmic implementations. Finally, as a-by-product, the inverse of the parametrized embedded normalizing flow, w.r.t. the estimated parameter values, serves as a surrogate model for the computational simulation model, which provides useful information for quantifying model discrepancies and sensitivity analysis.

Neural posterior estimation (NPE) enables Bayesian inference of the posterior parameter in agent-based models (ABMs) with intractable likelihoods, but simulating training data can be computationally expensive. We propose Double-Flow Neural Posterior Estimation (DF-NPE), which combines learned time-series summaries with two conditional normalizing flows for synthetic-data generation and posterior estimation. The framework supplements direct simulations with synthetic training pairs and supports both pooled and ordered training strategies. Experiments in two settings of the Brock-Hommes asset-pricing model show that DF-NPE can reduce direct simulations by 90 percent while achieving lower mean posterior discrepancies and runtime than standard NPE. Performance varies with the allocation of directly simulated and synthetic data and the training strategy. These results establish learned summary augmentation as a practical approach to reducing simulation costs for Bayesian posterior inference of parameters in ABMs.