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.
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.
Figure 2: Approximate ground-truth posteriors for the Brock-Hommes source and target models.
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.
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.
Figure 5: Approximate ground-truth posteriors for the MVGBM source and target models.
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.
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.
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.
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.
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
Figure 11: Posterior samples for SNPE from for the Brock-Hommes Model at β=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.
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
αw
η
σc
αw
η
σc
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.
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.
Junyao Zhang, Jinglai Li, Junqi Tang
School of Mathematics, University of Birmingham, Birmingham, United Kingdom
Agent-based modelling (ABM) is a widespread approach to simulate complex systems. Advancements in computational processing and storage have facilitated the adoption of ABMs across many fields; however, ABMs face challenges that limit their use as decision-support tools. A significant issue is parameter estimation in large-scale ABMs, particularly due to computational constraints on exploring the parameter space. This study evaluates a state-of-the-art simulation-based inference (SBI) framework that uses neural networks (NN) for parameter estimation. This framework is applied to an established labour market ABM based on job transition networks. The ABM is initiated with synthetic datasets and the real U.S. labour market. Next, we compare the effectiveness of summary statistics derived from a list of statistical measures with that learned by an embedded NN. The results demonstrate that the NN-based approach recovers the original parameters when evaluating posterior distributions across various dataset scales and improves efficiency compared to traditional Bayesian methods.
M Lopes Alves, Joel Dyer, Doyne Farmer +2
Department of Computer Science Institute of New Economic Thinking University of Oxford Oxford, UK · Department of Computer Science University of Oxford Oxford, UK
We present Causal Posterior Estimation (CPE), a novel method for Bayesian inference in simulator models, where evaluating the likelihood function is intractable or computationally expensive, but generating outputs given parameter values is straightforward. CPE approximates the posterior distribution using flow matching while directly incorporating the conditional dependence structure induced by the model's graphical representation into the neural network architecture. Across extensive experiments, we demonstrate that hard-coding these conditional dependencies into the network, rather than requiring them to be learned from data, enables CPE to achieve highly accurate posterior inference that matches or outperforms state-of-the-art baselines.
Simon Dirmeier, Antonietta Mira
Swiss Data Science Center, ETH Zurich, Switzerland · Università della Svizzera italiana, Switzerland · Insubria University, Italy