Amortized Bayesian Inference (ABI) enables efficient posterior estimation using generative neural networks trained on simulated data, but often suffers from performance degradation under model misspecification. While self-consistency (SC) training on unlabeled empirical data can enhance network robustness, current approaches are limited to static, single-task settings and fail to handle sequentially arriving data or distribution shifts. We propose a continual learning framework for ABI that decouples simulation-based pre-training from unsupervised sequential SC fine-tuning on real-world data. To address the challenge of catastrophic forgetting, we introduce two adaptation strategies: (1) SC with episodic replay, utilizing a memory buffer of past observations, and (2) SC with elastic weight consolidation, which regularizes updates to preserve task-critical parameters. Across three diverse case studies, our methods significantly mitigate forgetting and yield posterior estimates that outperform standard simulation-based training, achieving estimates closer to MCMC reference, providing a viable path for trustworthy ABI across a range of different tasks.
Figures & tables
Figure 1: Experiment 1. MMD ratio in log scale across CL tasks (0–9) for different methods. Boxes summarize variability across subsets. Dashed line marks parity with simulation-based (SB) baseline (ratio = 1). Naive SC shows catastrophic forgetting in CL setting whereas our proposed methods mitigate forgetting and provide better posterior estimates compared to SB and Naive SC. Test-time SC also gives accurate posterior estimates.
Figure 2: Experiment 2. MMD ratio in log scale for different methods aggregated over fifteen CL tasks. Dashed line marks parity with simulation-based baseline (ratio = 1). Naive SC shows catastrophic forgetting in CL setting whereas our proposed methods mitigate forgetting and perform better than both SB baseline and naive SC.
Figure 3: Experiment 3. MMD ratio in log scale across CL tasks (0–8) for the four RDM parameters for different methods. Boxes summarize variability over datasets. Dashed line marks parity with simulation-based baseline (ratio = 1). All SC methods outperform SB-only training. Catastrophic forgetting is not evident.
Appendix figures & tables13 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 4: Experiment 1. MMD ratio in log scale across CL tasks (0–9) for SC-EWC with different values of EWC hyperparameter λ . Boxes summarize variability across subsets. Dashed line marks parity with simulation-based (SB) baseline (ratio = 1). λ=102 shows forgetting while other variants perform similarly.
Figure 5: Experiment 1. MMD ratio in log scale across CL tasks (0–9) for SC-ER-EWC with different values of EWC hyperparameter λ . Boxes summarize variability across subsets. Dashed line marks parity with simulation-based (SB) baseline (ratio = 1). The performance remains largely stable even for smaller λ values, indicating that the robustness improvements are primarily attributable to episodic replay. The addition of EWC provides only marginal benefits in this setting.
Figure 6: Experiment 1. Absolute mean bias of linear regression parameters compared to Stan based estimates across CL tasks (0–9) for various methods. Boxes summarize variability across subsets. SB and naive SC show catastrophic forgetting while CL methods perform consistently better across tasks.
Figure 7: Experiment 1. Absolute standard deviation bias of linear regression parameters compared to Stan based estimates across CL tasks (0–9) for various methods. Boxes summarize variability across subsets.
Figure 8: Experiment 1. Absolute mean bias of linear regression parameter σ compared to Stan based estimates across CL tasks (0–9) for various methods. Boxes summarize variability across subsets. SB and naive SC show high bias across tasks while other methods perform consistently better.
Figure 9: Experiment 1. Absolute standard deviation bias of linear regression parameter σ compared to Stan based estimates across CL tasks (0–9) for various methods. Boxes summarize variability across subsets. SB shows high bias across tasks while other methods perform consistently better.
Figure 10: Experiment 2. Absolute mean bias of air traffic model parameters compared to Stan based estimates for various methods aggregated over 15 CL tasks. Boxes summarize variability of parameters across tasks. SB and naive SC show high bias across tasks while other methods perform consistently better.
Figure 11: Experiment 2. Absolute standard deviation bias of air traffic model parameters compared to Stan based estimates for various methods aggregated over 15 CL tasks. Boxes summarize variability of parameters across tasks. SB and naive SC show high bias across tasks while other methods perform consistently better.
Figure 12: Experiment 2. MMD ratio in log scale across CL tasks (1–15) for different methods. Dashed line marks parity with simulation-based (SB) baseline (ratio = 1). Naive SC shows catastrophic forgetting in CL setting whereas our proposed methods mitigate forgetting and provide better posterior estimates compared to SB and Naive SC. Test-time SC also gives accurate posterior estimates.
Figure 13: Experiment 2. MMD ratio in log scale aggregated over fifteen CL tasks for SC-ER with varying replay memory buffer K . Dashed line marks parity with simulation-based baseline (ratio = 1). For values of K≤6 the performance of SC-ER starts to decrease whereas for K≥7 , good performance is observed.
Figure 14: Experiment 2. MMD ratio in log scale aggregated over fifteen CL tasks for SC-ER-EWC with varying replay memory buffer K . Dashed line marks parity with simulation-based baseline (ratio = 1). For values of K=2,3 the performance of SC-ER-EWC decreases but still remain better than SB baseline as well as SC-ER for same K-value. It signifies that addition of EWC leads to enhanced performance when replay memory is limited.
Figure 15: Experiment 3. Absolute bias of the posterior mean of the marginal posteriors of the racing diffusion model.
Figure 16: Experiment 3. Absolute bias of the posterior standard deviation of the marginal posteriors of the racing diffusion model.
Continual learning is an online paradigm where a learner continually accumulates knowledge from different tasks encountered over sequential time steps. Importantly, the learner is required to extend and update its knowledge without forgetting about the learning experience acquired from the past, and while avoiding the need to retrain from scratch. Given its sequential nature and its resemblance to the way humans think, continual learning offers an opportunity to address several challenges which currently stand in the way of widening the range of applicability of deep models to further real-world problems. The continual need to update the learner with data arriving sequentially strikes inherent congruence between continual learning and Bayesian inference which provides a principal platform to keep updating the prior beliefs of a model given new data, without completely forgetting the knowledge acquired from the old data. This survey inspects different settings of Bayesian continual learning, namely task-incremental learning and class-incremental learning. We begin by discussing definitions of continual learning along with its Bayesian setting, as well as the links with related fields, such as domain adaptation, transfer learning and meta-learning. Afterwards, we introduce a taxonomy offering a comprehensive categorization of algorithms belonging to the Bayesian continual learning paradigm. Meanwhile, we analyze the state-of-the-art while zooming in on some of the most prominent Bayesian continual learning algorithms to date. Furthermore, we shed some light on links between continual learning and developmental psychology, and correspondingly introduce analogies between both fields. We follow that with a discussion of current challenges, and finally conclude with potential areas for future research on Bayesian continual learning.
Tameem Adel
NPL, Maxwell Centre, University of Cambridge, JJ Thomson Avenue, Cambridge, CB3 OHE, United Kingdom.
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.
Continual learning struggles to balance retaining past knowledge with absorbing new tasks. Stefan-CL elegantly resolves this stability-plasticity dilemma through the physics of melting. It frames consolidated knowledge as a protected "solid" and unused capacity as an adaptable "liquid." As the network learns, this boundary expands, governed by a "latent heat" tuning dial. By mathematically freezing the learned interior, Stefan-CL cuts forgetting to near zero, matching memory-heavy baselines without storing raw data, forging a beautiful, physics-grounded path for AI.