Adapting Neural Posterior Estimators for Simulation-Based Inference in Agent-Based Models

Junyao Zhang; Jinglai Li; Junqi Tang

cs.LGJan 12, 2026

Adapting Neural Posterior Estimators for Simulation-Based Inference in Agent-Based Models

Authors: Junyao Zhang, Jinglai Li, Junqi Tang

Organizations: School of Mathematics, University of Birmingham

Abstract

Agent-based models (ABMs) are widely used in economics and social sciences. They represent heterogeneous personal behaviors and their interactions by using simulations, whereas their likelihoods are often intractable for parameter estimations. Therefore, most traditional statistical inference techniques are hindered. Driven by recent advances in probabilistic machine learning, recent works have applied neural networks to achieve likelihood-free inference in large-scale ABMs. Specifically, Neural Posterior Estimation using normalizing flows provide amortized Bayesian inference from simulations, yet a pre-trained estimator may become inaccurate when the target task changes. In this work, we investigate the practicality of adapting a pre-trained conditional normalizing flow through a unified sequential neural posterior estimation test-time training (SNPE-TTT) framework. The framework includes four adaptation strategies: full-parameter adaptation, low-rank adaptation, and two update spaces identified from target-task gradients. We evaluate all four methods across economic simulation models with both tractable and intractable likelihoods. These benchmarks demonstrate that the sequential neural posterior estimation test-time training framework can adapt pre-trained posterior estimators to related inference tasks using a limited budget of target simulations. Posterior recovery and computational benefits depend on the task and adaptation strategy, with lower target-stage cost in some settings but no consistently superior posterior accuracy.

Figures & tables

\mathcal{D}:=\mathcal{D}\cup\bigcup_{n=1}^{N}\{(x_{n}^{(m)},\theta_{n}^{(m)})\}.

Algorithm 1 Sequential Neural Posterior Estimation

Panel of Figure 1 — Figure 1: The SNPE-TTT framework. A source-trained posterior estimator is adapted using sequentially acquired target simulations. The selected strategy determines the trainable parameters, update constraints and basis-refresh schedule.

Panel of Figure 2 — Figure 2: Approximate ground-truth posteriors for the Brock-Hommes source and target models.

Panel of Figure 3 — Figure 3: Posterior samples from the four SNPE-TTT strategies for the Brock–Hommes. Diagonal panels show marginal posterior distribution (blue curve), upper-triangular panels show bivariate joint density. Red lines and points denote the mean of the likelihood-based reference posterior. Darker (resp. lighter) colours indicate lower (resp. higher) posterior density in off-diagonal.

Panel of Figure 4 — Figure 4: Boxplots of Brock-Hommes discrepancies over 20 repeated trails for each approach.

Method	WASS, median [IQR]	MMD 2 , median [IQR]
SNPE-FPA	0.1364 [0.1207, 0.1631]	0.5199 [0.4429, 0.5604]
SNPE-LoRA	0.2939 [0.2479, 0.4429]	0.6537 [0.6016, 0.6846]
GradSubspace-FPA	0.1838 [0.1558, 0.2216]	0.5133 [0.4808, 0.5307]
GradSubspace-PEA	0.1789 [0.1560, 0.2072]	0.4905 [0.4560, 0.5272]

Table 1: Brock-Hommes posterior discrepancies across 20 repeated trails. Bold values mark the lowest descriptive median in each column.

Panel of Figure 5 — Figure 5: Approximate ground-truth posteriors for the MVGBM source and target models.

Panel of Figure 6 — Figure 6: Posterior samples from the four SNPE-TTT strategies for the MVGBM. Diagonal panels show marginal posterior distribution (blue curve), upper-triangular panels show bivariate joint density. Red lines and points denote the mean of the likelihood-based reference posterior. Darker (resp. lighter) colours indicate lower (resp. higher) posterior density in off-diagonal.

Panel of Figure 7 — Figure 7: Boxplots of MVGBM discrepancies over 20 repeated trails for each approach.

Method	WASS, median [IQR]	MMD 2 , median [IQR]
SNPE-FPA	0.2221 [0.1964, 0.2611]	0.0715 [0.0592, 0.0908]
SNPE-LoRA	0.2825 [0.2345, 0.3672]	0.1053 [0.0874, 0.1470]
GradSubspace-FPA	0.2137 [0.1934, 0.2278]	0.0604 [0.0510, 0.0708]
GradSubspace-PEA	0.2169 [0.1898, 0.2284]	0.0598 [0.0511, 0.0742]

Table 2: MVGBM posterior discrepancies across 20 repeated trails. Bold values mark the lowest descriptive median in each column.

Panel of Figure 8 — Figure 8: Posterior samples from SNPE-TTT for the source (left) and target (right) Franke-Westerhoff Model. Diagonal panels show marginal posterior distribution (blue curve), upper-triangular panels show bivariate joint density. Red lines and points indicate the generating parameter values. Darker (resp. lighter) colours indicate lower (resp. higher) posterior density in off-diagonal.

Panel of Figure 9 — Figure 9: Posterior samples from the four SNPE-TTT strategies for the Franke-Westerhoff Model. Diagonal panels show marginal posterior distribution (blue curve), upper-triangular panels show bivariate joint density. Red lines and points indicate the generating parameter values. Darker (resp. lighter) colours indicate lower (resp. higher) posterior density in off-diagonal.

Panel of Figure 10 — Figure 10: Rank histograms generated according to the simulation-based calibration for SNPE-FPA, SNPE-LoRA, GradSubspace-FPA and GradSubspace-PEA in posterior inference of FW model, respectively. The shaded regions represent the symmetric 95% point function of a Binomial random variable with success probability 1/P=1/10, that is, the bottom (resp. top) of the shaded regions is the inverse of the cumulative distribution function for a Binomial random variable at quantile 0.025 (resp. 0.975). This generally stands for the expected deviations from complete uniformity when the histograms are distributed uniformly.

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 11 — Figure 11: Posterior samples for SNPE from for the Brock-Hommes Model at $\beta=60$ (left) and target MVGBM (right) in Section 4 , which use [500, 500, 500, 1000] as simulation budgets. Diagonal panels show marginal posterior distribution (blue curve), upper-triangular panels show bivariate joint density. Red lines and points denote the mean of the likelihood-based reference posterior. Darker (resp. lighter) colours indicate lower (resp. higher) posterior density in off-diagonal.

Panel of Figure 12 — Figure 12: Rank histograms generated according to the simulation-based calibration for source (top) and target (bottom) FW model. The shaded regions represent the symmetric 95% point function of a Binomial random variable with success probability 1/P=1/10, that is, the bottom (resp. top) of the shaded regions is the inverse of the cumulative distribution function for a Binomial random variable at quantile 0.025 (resp. 0.975). This generally stands for the expected deviations from complete uniformity when the histograms are distributed uniformly.

		90% coverage (%)			Mean normalised width
Method	Cases	$\alpha_{w}$	$\eta$	$\sigma_{c}$	$\alpha_{w}$	$\eta$	$\sigma_{c}$
Source-only	77	87.0	92.2	83.1	0.875	0.742	0.284
Scratch SNPE	95	86.3	91.6	89.5	0.886	0.676	0.245
SNPE-FPA	98	85.7	92.9	88.8	0.871	0.615	0.225
SNPE-LoRA	99	86.9	88.9	92.9	0.866	0.614	0.236
GradSubspace-FPA	98	84.7	91.8	89.8	0.869	0.665	0.253

Appendix

Table 3: FW marginal 90% coverage and interval widths, normalised by the prior range, on completed observations.

		Contrast
Model	Metric	G-FPA $-$ FPA	PEA $-$ FPA	PEA $-$ G-FPA
BH	WASS: Holm $p$	0.000629	0.002865	0.621513
	Median difference	0.041138	0.036054	0.001398
	MMD 2 : Holm $p$	1.000000	1.000000	1.000000
	Median difference	$-0.007552$	0.015908	$-0.009274$
MVGBM	WASS: Holm $p$	0.404900	0.398180	0.404900

Appendix

Table 4: Complete paired comparisons (

n=20

each). FPA, G-FPA and PEA denote SNPE-FPA, GradSubspace-FPA and GradSubspace-PEA. Holm correction is separate for each model–metric family.

Simulation-Based Inference

Simulation-based inference estimates parameters and uncertainty from simulators when evaluating a tractable likelihood is difficult or impossible.

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Simulation-Efficient Double-Flow Neural Posterior Estimation for Agent-Based Models

Junyao Zhang, Jinglai Li, Junqi Tang

School of Mathematics, University of Birmingham, Birmingham, United Kingdom

Neural Posterior Estimation Simulation-Based Inference Agent-Based Modeling Flow-Based Generative Modeling

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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.

Abstract

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