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

Junyao Zhang; Jinglai Li; Junqi Tang

cs.MAOct 26, 2026

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

Authors: Junyao Zhang, Jinglai Li, Junqi Tang

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

Abstract

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.

Figures & tables

Panel of Figure 1 — Figure 1: DF-NPE with three posterior-training strategies. (A) Direct simulations train the encoder and latent-surrogate flow; stored trajectories are re-encoded, and the current surrogate generates synthetic summaries from retained parameter proposals. (B) The posterior flow is trained using Mixed, Synthetic-to-Simulation or Simulation-to-Synthetic, with the encoder and surrogate held fixed. Blue and teal denote directly-simulated and synthetic data, respectively. The dashed arrow indicates that the fitted posterior, conditioned on the observed summary, supplies the proposal for the next round.

Panel of Figure 2 — Figure 2: Posterior samples from the approximate ground truth, SNPE and DF-NPE for the smooth Brock–Hommes setting. DF-NPE uses Mixed training at a directly-simulated:synthetic ratio of 3:7. Diagonal panels show marginal posterior distributions as count histograms; upper-triangular panels show bivariate joint distributions as two-dimensional count histograms. Red lines and points denote the mean of the likelihood-based reference posterior. Darker and lighter colours indicate lower and higher sample counts, respectively, in the off-diagonal plots.

Panel of Figure 3 — Figure 3: Posterior discrepancies of SNPE and DF-NPE with Mixed training for the smooth Brock–Hommes setting. Results show WASS and MMD 2 over ten fits per configuration using one fixed observation. Ratios denote directly-simulated:synthetic allocations; SNPE uses only direct simulations. White lines denote medians, boxes show the interquartile range, and whiskers extend to the most extreme values within 1.5 interquartile ranges.

Panel of Figure 4 — Figure 4: Posterior samples from the approximate ground truth, SNPE and DF-NPE for the noisy Brock–Hommes setting. DF-NPE uses Mixed training at a directly-simulated:synthetic ratio of 3:7. Diagonal panels show marginal posterior distributions as count histograms; upper-triangular panels show bivariate joint distributions as two-dimensional count histograms. Red lines and points denote the mean of the likelihood-based reference posterior. Darker and lighter colours indicate lower and higher sample counts, respectively, in the off-diagonal plots.

Panel of Figure 5 — Figure 5: Posterior discrepancies of SNPE and DF-NPE with Mixed training for the noisy Brock–Hommes setting. Results show WASS and MMD 2 over ten fits per configuration using one fixed observation. Ratios denote directly-simulated:synthetic allocations; SNPE uses only direct simulations. White lines denote medians, boxes show the interquartile range, and whiskers extend to the most extreme values within 1.5 interquartile ranges.

Panel of Figure 6 — Figure 6: Posterior discrepancies of the three DF-NPE training strategies and SNPE for the smooth Brock–Hommes setting. Results show WASS and MMD 2 over ten fits at directly-simulated:synthetic ratios of 1:9, 3:7 and 5:5. Syn. $\to$ Sim. and Sim. $\to$ Syn. denote Synthetic-to-Simulation and Simulation-to-Synthetic, respectively. White lines denote medians, boxes show the interquartile range, and whiskers extend to the most extreme values within 1.5 interquartile ranges.

Panel of Figure 7 — Figure 7: Posterior discrepancies of the three DF-NPE training strategies and SNPE for the noisy Brock–Hommes setting. Results show WASS and MMD 2 over ten fits at directly-simulated:synthetic ratios of 1:9, 3:7 and 5:5. Syn. $\to$ Sim. and Sim. $\to$ Syn. denote Synthetic-to-Simulation and Simulation-to-Synthetic, respectively. White lines denote medians, boxes show the interquartile range, and whiskers extend to the most extreme values within 1.5 interquartile ranges.

Panel of Figure 8 — Figure 8: End-to-end runtime of SNPE and the three DF-NPE training strategies for the smooth and noisy Brock–Hommes settings. Results show ten runs per configuration at directly-simulated:synthetic ratios of 1:9, 3:7 and 5:5, with runtime measured in minutes. White lines denote medians, boxes show the interquartile range, and whiskers extend to the most extreme values within 1.5 interquartile ranges.

Appendix figures & tables9 assets

Supplementary material from the paper’s appendix.

Appendix

Component	Setting
Supervised encoder	Two-layer GRU, hidden width 32 and latent dimension 8; parameter-regression head with one 32-unit hidden layer
Encoder fit	Adam, learning rate $10^{-3}$ , batch size 100, at most 100 epochs; latent extraction batch size 256
Conditional latent surrogate	Five-transform MAF with 50 hidden features; learning rate $5\times 10^{-4}$ , batch size 100, 10% validation and early stopping after 10 validation epochs
Latent posterior estimator	Five-transform SNPE-C MAF with 50 hidden features; learning rate $5\times 10^{-4}$ , batch size 200, 10% validation and early stopping after 20 validation epochs
SNPE comparator	Jointly trained two-layer GRU embedding and MAF posterior through the project defaults
Execution	Timing runs capped neural fits at 500 epochs

Appendix

Table A.1 : Architecture and optimisation settings for Mixed and SNPE. Sequential budgets and allocations are specified in Section 3.2.1 .

Workflow	90% coverage	Median RMSE	Summary MAE
1:9	$0.979\pm 0.020$	$0.577\pm 0.294$	$0.380\pm 0.318$
3:7	$0.931\pm 0.093$	$0.459\pm 0.074$	$0.267\pm 0.105$
5:5	$0.957\pm 0.050$	$0.513\pm 0.060$	$0.190\pm 0.027$
7:3	$0.951\pm 0.097$	$0.482\pm 0.089$	$0.227\pm 0.063$
9:1	$0.973\pm 0.022$	$0.484\pm 0.083$	$0.203\pm 0.049$
SNPE	$0.932\pm 0.011$	$0.372\pm 0.015$	$0.222\pm 0.014$

Appendix

Table B.1: Smooth Brock–Hommes: posterior-predictive checks over ten fits, using 200 posterior draws per fit returned to the simulator. Entries report mean

\pm

sample standard deviation.

Panel of Figure 11 — Figure C.1 : Posterior samples from the two ordered DF-NPE training strategies for the smooth and noisy Brock–Hommes settings at a directly-simulated:synthetic ratio of 3:7. Syn. $\to$ Sim. and Sim. $\to$ Syn. denote Synthetic-to-Simulation and Simulation-to-Synthetic, respectively. Diagonal panels show marginal posterior distributions as count histograms; upper-triangular panels show bivariate joint distributions as two-dimensional count histograms. Red lines and points denote the mean of the likelihood-based reference posterior. Darker and lighter colours indicate lower and higher sample counts, respectively, in the off-diagonal plots.

Ratio	Direct/round	WASS gain	MMD 2 gain	MAE gain
1:9	100	$0.202\pm 0.140$ (5/5)	$0.295\pm 0.308$ (3/5)	$0.083\pm 0.049$ (5/5)
3:7	300	$0.216\pm 0.096$ (5/5)	$0.403\pm 0.195$ (5/5)	$0.067\pm 0.027$ (5/5)
5:5	500	$0.229\pm 0.172$ (5/5)	$0.465\pm 0.347$ (5/5)	$0.070\pm 0.054$ (5/5)
7:3	700	$0.260\pm 0.063$ (5/5)	$0.484\pm 0.150$ (5/5)	$0.076\pm 0.027$ (5/5)
9:1	900	$0.310\pm 0.057$ (5/5)	$0.573\pm 0.169$ (5/5)	$0.087\pm 0.015$ (5/5)

Appendix

Table C.1: Smooth Brock–Hommes: same-direct-budget comparisons over five paired seeds. Entries report mean

\pm

sample standard deviation of SNPE-minus-Mixed discrepancies; positive values favour Mixed. Parentheses give improving seeds out of five.

Smooth Brock–Hommes
Workflow	WASS	MMD 2	MAE to $\theta^{*}$	Time (min)	Speed-up
1:9	$0.3051\pm 0.0841$	$0.5637\pm 0.1997$	$0.0780\pm 0.0256$	$10.15\pm 2.21$	$6.45\times$
3:7	$0.2681\pm 0.1042$	$0.4595\pm 0.2168$	$0.0685\pm 0.0326$	$16.03\pm 5.59$	$4.08\times$
5:5	$0.2458\pm 0.1228$	$0.4007\pm 0.2635$	$0.0590\pm 0.0382$	$19.13\pm 2.91$	$3.42\times$
7:3	$0.2489\pm 0.1239$	$0.4478\pm 0.2968$	$0.0656\pm 0.0414$	$21.47\pm 2.14$	$3.05\times$
9:1	$0.2403\pm 0.1027$	$0.4430\pm 0.2559$	$0.0622\pm 0.0332$	$25.63\pm 1.11$	$2.55\times$

Appendix

Table C.2: Mixed workflows and SNPE across the two Brock–Hommes settings. Entries are mean

\pm

sample standard deviation over ten seeds. Scores and timing conventions follow Section 3.1 . Ratios are directly-simulated:synthetic allocations; time is in minutes.

Figure C.2 : Computational cost of SNPE and DF-NPE with Mixed training for the Brock–Hommes model, broken down by workflow phase. Stacked segments show median phase times in minutes over ten seeds; ratios denote directly-simulated:synthetic allocations. Colours identify the computational phases. The setup / I/O / plotting segment contains costs not assigned to the other phases. Segment medians need not sum to the median end-to-end runtime.

Smooth Brock–Hommes
Ratio	Schedule	WASS	MMD 2	MAE to $\theta^{*}$	Time (min)
1:9	Syn. $\to$ Sim.	$0.4182\pm 0.1407$	$0.76124\pm 0.24726$	$0.1223\pm 0.0406$	$27.71\pm 26.66$
1:9	Sim. $\to$ Syn.	$0.3696\pm 0.1209$	$0.67745\pm 0.27146$	$0.1017\pm 0.0376$	$17.70\pm 9.69$
3:7	Syn. $\to$ Sim.	$0.2993\pm 0.0888$	$0.55649\pm 0.23965$	$0.0804\pm 0.0258$	$15.39\pm 7.99$
3:7	Sim. $\to$ Syn.	$0.3822\pm 0.1739$	$0.76860\pm 0.47356$	$0.1011\pm 0.0463$	$17.22\pm 5.87$
5:5	Syn. $\to$ Sim.	$0.2612\pm 0.1182$	$0.44882\pm 0.31734$	$0.0676\pm 0.0336$	$25.23\pm 20.61$

Appendix

Table C.3 : Complete ordered-workflow results across ten seeds (mean

\pm

sample standard deviation). Syn.

\to

Sim. denotes Synthetic-to-Simulation and Sim.

\to

Syn. denotes Simulation-to-Synthetic. Scores follow Section 3.1 ; time is recorded end-to-end minutes. Mixed and SNPE results are in Table C.2 .

Task	Source	Ratio	Prior excess MMD 2	Ref. excess MMD 2
Smooth BH	$G\to S$	1:9	$0.36259\pm 0.03831$	$0.29850\pm 0.07948$
Smooth BH	$G\to S$	3:7	$0.22473\pm 0.04098$	$0.15527\pm 0.06224$
Smooth BH	$G\to S$	5:5	$0.20284\pm 0.03971$	$0.09657\pm 0.02512$
Smooth BH	$S\to G$	1:9	$0.27270\pm 0.05115$	$0.20251\pm 0.09400$
Smooth BH	$S\to G$	3:7	$0.25792\pm 0.04028$	$0.13531\pm 0.04036$
Smooth BH	$S\to G$	5:5	$0.21534\pm 0.04260$	$0.08688\pm 0.03532$

Appendix

Table D.1: Conditional surrogate diagnostics over ten source fits (mean

\pm

source-seed standard deviation). Prior and reference excess MMD 2 average ten parameter sites per region, with source-specific standardisation.

Task	Source	Ratio	$\Delta\mathrm{MMD}_{\mathrm{diag}}^{2}$	$\Delta W_{\mathrm{marg}}$	$\Delta e_{\mathrm{ref}}$
Smooth BH	$G\to S$	1:9	$-0.01428\pm 0.01305$ (9)	$-0.00933\pm 0.01407$ (7)	$-0.00989\pm 0.01472$ (7)
Smooth BH	$G\to S$	3:7	$-0.00624\pm 0.02317$ (5)	$-0.00832\pm 0.00733$ (9)	$-0.01047\pm 0.00958$ (8)
Smooth BH	$G\to S$	5:5	$-0.01409\pm 0.01681$ (7)	$-0.01087\pm 0.00864$ (9)	$-0.01311\pm 0.00914$ (10)
Smooth BH	$S\to G$	1:9	$-0.00143\pm 0.02758$ (7)	$-0.00077\pm 0.01900$ (5)	$-0.00365\pm 0.01970$ (5)
Smooth BH	$S\to G$	3:7	$-0.00139\pm 0.02299$ (6)	$0.00082\pm 0.02088$ (4)	$0.00044\pm 0.02295$ (5)
Smooth BH	$S\to G$	5:5	$-0.02537\pm 0.01602$ (9)	$-0.01076\pm 0.01356$ (8)	$-0.01327\pm 0.01485$ (8)

Appendix

Table D.2 : Frozen-component prior-only controls: augmented-minus-direct differences over ten source fits (mean

\pm

sample standard deviation). Parentheses give improvements out of ten; negative values favour augmentation.

\Delta\mathrm{MMD}_{\mathrm{diag}}^{2}

uses the standardised diagnostic kernel,

\Delta W_{\mathrm{marg}}

is prior-width-scaled marginal Wasserstein distance, and

\Delta e_{\mathrm{ref}}

is reference-mean error. Source labels identify the encoder–surrogate origin (

G\to S

: Synthetic-to-Simulation;

S\to G

: Simulation-to-Synthetic). Both augmented arms use pooled training. Both arms share the directly-simulated bank, initialisation and 1,000-update budget, but direct exposure differs. Diagnostic scores use different definitions from the main scores.

Neural Posterior Estimation

Neural posterior estimation learns probability distributions over unknown model parameters from observed data, often using simulations when likelihoods are unavailable.

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

Junyao Zhang, Jinglai Li, Junqi Tang

School of Mathematics, University of Birmingham

Simulation-Based Inference Neural Posterior Estimation Test-Time Training Agent-Based Modeling

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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.

Abstract

Neural Posterior Estimation

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