Simulation-based inference stops training only the posterior net: summary statistics take the lead, and training size grows only polynomially in observation dimension

Simulation-based inference (SBI) trains neural networks on simulated data to approximate posteriors when the likelihood is intractable; the new framework treats learning a low-dimensional summary statistic as the central object, rather than as an accessory to the posterior estimator. Earlier approaches either used neural approximations of the posterior mean as summaries, or, as in BayesFlow, learned a global amortized posterior estimator with the summary network mainly structuring that estimator. On a toy benchmark with a four-mode symmetric posterior, after mapping each parameter onto the sphere by inverse stereographic projection, summary-based methods achieved lower MMD and sliced Wasserstein errors than the observation-wise baselines NLE and TD-NPSE; SNPE-C without a summary performed poorly on these high-dimensional observations.

The summary map and the inference network are trained jointly: the summary network compresses high-dimensional data into a low-dimensional vector, and the inference network reconstructs the posterior over parameters from that vector, with training minimizing the discrepancy between the full-data posterior and the summary-conditioned posterior. A KL objective requires the inference network to provide tractable, normalized conditional densities, for example an invertible normalizing flow; with maximum mean discrepancy (MMD) or sliced MMD, the objective can be estimated directly from samples generated by the inference network, so the network need not have a tractable density. Sliced MMD replaces high-dimensional kernel evaluations with univariate computations along random one-dimensional projections, and with a fixed kernel bandwidth the number of samples needed for an informative gradient grows only linearly with the parameter dimension; on the SIR, Lotka–Volterra and radio-propagation benchmarks, ABC–MCMC local refinement after amortized training further reduced bias.

The work addresses researchers and engineers who need Bayesian inference when the likelihood is intractable, in settings where a simulator is available, observations are high-dimensional, and amortized inference can be reused. The theory covers i.i.d. data: exponential-family models with additive sufficient statistics, and a large-sample regime where the Bernstein–von Mises theorem makes the posterior approximately Gaussian. Experiments cover a four-mode toy benchmark, SIR, Lotka–Volterra, and a real radio-propagation dataset.

The large-sample corollary requires the sample size to be sufficiently large, and the attainable tolerance is of order root-n rather than arbitrary, because general models lack low-dimensional sufficient statistics and the gap between the true posterior and its Gaussian limit is irreducible. Local refinement introduces additional error from the positive ABC tolerance and finite Monte Carlo sampling, and refined posteriors are often conservative with some overcoverage. The guarantees are stated for i.i.d. data, with corresponding guarantees for dependent observations left for future work; using Bayesian synthetic likelihood instead of ABC as the refinement target would require the learned summary to be approximately Gaussian conditional on the parameter, a property the current objective does not guarantee.