Bayesian simulation-based inference (SBI) can struggle to recover increasingly concentrated posteriors as observation size grows. We consider learning sufficient summary statistics jointly with the posterior as a general principle for scalable inference in various SBI methods. A common measure taken in SBI is dimension reduction through predetermined summaries, which may partially discard posterior information. Our approach instead imposes structure on how the posterior learner depends on a learned summary (e.g., affinity), and learns the summary simultaneously with the posterior. We identify an important mechanism named two-way guidance: under a structural restriction on the summary, the posterior sufficiency of the learned summary simplifies posterior learning, whereas a suitable posterior objective guides the learned summary towards posterior sufficiency. This mechanism explains a counterintuitive finding that the structural restriction on the summary statistics improves posterior learning, rather than limit its performance. We establish theoretical guarantees on both posterior estimation error and the learned summary's departure from sufficiency, thus identifying conditions on the simulation budget for recovering increasingly concentrated posteriors as observation size grows.
Figures & tables
Figure 1: Gaussian posterior draws and contours, with the oracle posterior in the first row and increasing n across columns. All panels use common (μ,σ2) axes.
Figure 2: Gaussian posterior error: mean and SD across three seeds. Solid: summaries; dashed: raw data.
Figure 3: Brock–Hommes energy score: mean and SD across three seeds. Solid: summaries; dashed: raw data.
Appendix figures & tables2 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 4: Brock–Hommes posterior draws and marginal KDE contours: (a) (g2,b2) and (b) (g3,b3) . Rows show the five summary-based learners followed by their raw-data counterparts; columns within each half increase n . All panels use the first observed series, training seed 0 , and 2,000 posterior draws. Solid and dashed contours enclose 50% and 90% of the full fitted marginal KDE; stars mark the generating parameter. Axes are shared within each half, and some contours extend outside the displayed range.
Figure 5: Brock–Hommes energy scores for NPE and NRE on the same first observed series: mean and sample SD across three training seeds, using 2,000 posterior draws per run. Solid lines with filled markers denote learned summaries; dashed lines with hollow markers denote raw observations. Lower scores are better, and error bars show SD rather than confidence intervals.
Department of Statistics University of Washington · Department of Computer Science Cornell University · School of Mathematical Sciences and Center for Statistical Science Peking University +1