Simulation-Based Inference

Simulation-based inference with neural posterior estimation (NPE) often yields overconfident and unreliable posteriors under limited simulation budgets. To address this, we propose DRO-NPE, a distributionally robust approach that replaces the standard NPE objective with a worst-case loss over a Wasserstein ambiguity set. We introduce KL-based metrics for miscoverage and miscalibration, and use these to show that the DRO-NPE objective controls overfitting and reduces posterior overconfidence. Our method is tractable, parallelisable, and readily integrates with standard normalising flows. Across benchmark SBI tasks, DRO-NPE consistently improves coverage and calibration, while narrowing the gap between empirical and population NPE loss, leading to more reliable inference in low-simulation regimes.

Flow and diffusion generative models have established themselves as widely adopted density estimators for simulation-based inference (SBI), extending naturally from neural posterior estimation to likelihood and joint density estimation. Their principled optimization objectives and freedom from architectural constraints have driven rapid adoption across the natural sciences. Yet the most widely used SBI libraries remain PyTorch-based, leaving researchers who develop their forward models and analysis pipelines in JAX without a native option. We present GenSBI, an open-source library that implements flow matching, score matching, and denoising diffusion entirely in JAX. The library offers three transformer-based architectures - SimFormer, Flux1, and a novel Flux1Joint that extends gate-modulated transformer blocks to joint density estimation - all interchangeable through a unified interface that decouples generative method, neural backbone, and inference mode. GenSBI provides an end-to-end workflow from training through posterior calibration (SBC, TARP, LC2ST) and supports custom architectures with domain-specific embedding networks. We validate the framework on standard SBI benchmarks, achieving near-ideal mean C2ST scores (0.50-0.56, where 0.50 is ideal) on SBIBM tasks with minimal per-task tuning and well-calibrated posterior coverage across all tested configurations. The code is publicly available at https://github.com/aurelio-amerio/GenSBI.

State-space models (SSMs) are powerful probabilistic tools for modeling time-varying systems with latent dynamics. Inference in SSMs involves the estimation of latent states and parameters. In this work, we focus on parameter inference, which for SSMs is in general a very challenging problem due to the intractability of the likelihood. Recently, neural estimation methods, such as sequential neural likelihood (SNL), have shown promising results in Bayesian inference problems. In this paper, we show that SNL, when applied to the SSM setting, suffers important limitations, such as requiring a large amount of simulated samples to achieve a moderate performance, scaling poorly with sequence length, while not being amortized. We then introduce a novel inference algorithm called truncated-SNL (T-SNL), which addresses the limitations of SNL. Our algorithm is more accurate, more stable and robust during training, more scalable to longer temporal sequences, and can be amortized when new observations become available. Our experiments show that T-SNL is sample-efficient, robust, and flexible algorithm which outperforms other approaches.

Compositional score-based approaches to simulation-based inference (SBI) approximate the posterior over a shared parameter given $n$ independent observations by aggregating individually learned posterior scores: currently, there are two main propositions of such methods (Geffner et al. (2023), Linhart et al. (2026)). As the resulting composite score does not correspond to the score of any distribution along the forward diffusion path of the true multi-observation posterior, sampling from it via a reverse SDE leads to an irreducible bias. Annealed Langevin dynamics provides a principled alternative: it treats the composite score as the genuine score of a sequence of tractable bridging densities and samples from them in succession. When properly tuned, it could lead to a controllable bias. However, its hyperparameters, namely step sizes, the number of steps per level, and the number of annealing levels, have so far been chosen empirically. We derive Wasserstein bounds for annealed Langevin with approximate scores and translate them into explicit decision rules for these hyperparameters that guarantee a prescribed sampling accuracy, while highlighting different theoretical aspects of each composite score formulation. In the Gaussian setting, we obtain closed-form expressions for all relevant quantities and prove that the bridging densities of Linhart et al. (2026) consistently admit larger step sizes and require fewer total Langevin steps than those of Geffner et al. (2023). Furthermore, we show empirically that the tuning obtained in the Gaussian setting generalizes to more complex problems, thus providing a well-understood and theoretically grounded starting point for practitioners using compositional score-based approaches.

Neural Posterior Estimation (NPE) enables rapid parameter inference for complex simulators with intractable likelihoods. NPE trains an inference network to estimate a probability density over parameters given data, typically assumed to be \emph{continuous}. However, many scientific models involve parameter spaces that are \emph{mixed}, that is, they contain both discrete and continuous dimensions. We address this limitation by extending NPE to mixed parameter spaces through an inference network that jointly handles discrete and continuous parameters. The inference network factorizes the joint posterior into discrete and continuous components, combining an autoregressive classifier for the discrete parameters with a generative model for the continuous parameters, trained jointly under a single simulation-based objective. In addition, we propose a diagnostic tool to assess the calibration of the mixed posterior approximation. Across tractable toy examples and real-world scientific simulators, our joint inference approach yields accurate and calibrated posteriors. The inference framework is available in the \texttt{sbi} Python package.

A likelihood-free transport filtering method is proposed based on the couplings between state and observation variables. By exploiting a block-triangular structure in the transport map, the analysis step of filtering is reformulated as the minimization of the maximum mean discrepancy (MMD) between the true joint measure and its transport-based approximation. To circumvent the non-convexity in the MMD optimization, we introduce a training-free transport filter method via gradient flows, which leads to an analytic computation for the transport map that implies the steepest descent direction of the MMD. The proposed approach accurately approximates non-Gaussian filtering posteriors and avoids particle collapse. We provide a convergence analysis for the expectation of the MMD between the approximated posterior and the truth posterior. Finally, we extend the method to high-dimensional problems through domain localization. Numerical examples demonstrate the superior performance of our approach over conventional filtering methods in nonlinear, non-Gaussian scenarios.

Neural likelihood surrogates (e.g., Neural Ratio Estimation) for stochastic process models are commonly trained via probabilistic classification on simulated data, which forces a tradeoff between surrogate quality and training costs. For structured models where the exact score $\nabla_θ\log p(x \mid θ)$ is available, this information can be incorporated into training by augmenting the cross-entropy loss with a score-matching term. However, the optimal weighting of the two losses is not known a priori, and selecting it by hand requires expensive tuning that undercuts the computational savings. We propose an adaptive, tuning-free algorithm that sets the score loss weights during training based on loss gradients, adding minimal overhead to standard classifier training. We evaluate our approach on case studies involving network dynamics and spatial processes, demonstrating that it improves surrogate quality at a drastically lower computational cost than generating more training data. Notably, in some cases, our approach achieves downstream inference performance equivalent to a 10x increase in training data with less than a 1.1x increase in training time.

In this work, we study TabPFN as a training-free, modular summary network for simulation-based Bayesian inference (SBI). Tabular foundation models such as TabPFN are pretrained on broad families of synthetic tabular data-generating processes and adapt at test time through in-context learning, making them natural candidates for SBI, where posterior estimation often depends on learning informative summaries of simulated observations. We propose PFN-NPE: a general recipe that uses a pretrained TabPFN encoder as a fixed summary network for simulator outputs, then pairs the resulting summaries with a downstream inference head chosen for the problem. With normalizing flows as the default inference head, PFN-NPE matches established posterior approximation methods and sometimes outperforms them. More importantly, diagnostic probes show that the TabPFN-derived summaries often preserve useful posterior location and marginal information. These analyses also reveal a limitation in that TabPFN-derived summaries may struggle to represent the joint posterior structure even when the marginals are well recovered. Still, our experiments show that TabPFN can serve as an effective summary network across a diverse set of SBI settings, with the inference network left modular and task-dependent.

Simulation-based inference (SBI) provides amortized Bayesian parameter inference from simulator-generated data without requiring explicit likelihood evaluation. Its reliability can degrade under model misspecification, where real-world observations are not well represented by the simulator used for training. Existing methods using unlabeled real-world data often align simulated and real-world data distributions, but marginal alignment alone does not directly preserve parameter-relevant information needed for posterior inference. We propose SPIN, an SBI framework with parameter-relevant information-preserving domain transfer using unlabeled, unpaired real-world observations. During training, SPIN translates labeled simulator observations toward the real-world domain and back to the simulator domain, using the original simulator labels to encourage domain transfer that preserves parameter-relevant mutual information. At test time, the learned real-to-simulator transport maps real-world observations into the simulator domain for posterior inference, without requiring real-world parameter labels or paired real--simulator observations. Across controlled synthetic and physical real-world benchmarks, SPIN improves real-world posterior inference, with the improvement becoming clearer as misspecification increases.

Radar sounders are electromagnetic instruments that can probe deep into the subsurface of Earth and other planetary bodies by processing the echo of transmitted radar waves. Conventional approaches for analyzing such data rely on approximate assumptions and often produce point estimates that ignore parameter correlations as well as galactic and measurement noise. We propose a simulation-based inference approach to terrain parameter inversion from radar sounder data, where synthetic observations from a GPU-based simulator are used to train a neural network-based density estimator for neural posterior estimation (NPE). By explicitly conditioning on reference surface assumptions, the proposed framework allows systematic evaluation of posterior robustness to reference surface variability. We demonstrate that our NPE model is well calibrated on simulated data and transferable to real Mars radar profiles, where we analyze terrain parameters using literature-informed reference values.

Accurate condition monitoring of industrial equipment requires inferring latent degradation parameters from indirect sensor measurements under uncertainty. While traditional Bayesian methods like Markov Chain Monte Carlo (MCMC) provide rigorous uncertainty quantification, their heavy computational bottlenecks render them impractical for real-time process control. To overcome this limitation, we propose an AI-driven framework utilizing Simulation-Based Inference (SBI) powered by amortized neural posterior estimation to diagnose complex failure modes in heat exchangers. By training neural density estimators on a simulated dataset, our approach learns a direct, likelihood-free mapping from thermal-fluid observations to the full posterior distribution of degradation parameters. We benchmark this framework against an MCMC baseline across various synthetic fouling and leakage scenarios, including challenging low-probability, sparse-event failures. The results show that SBI achieves comparable diagnostic accuracy and reliable uncertainty quantification, while accelerating inference time by a factor of82$\times$ compared to traditional sampling. The amortized nature of the neural network enables near-instantaneous inference, establishing SBI as a highly scalable, real-time alternative for probabilistic fault diagnosis and digital twin realization in complex engineering systems.

The cost of simulator evaluations is a key practical bottleneck for Simulation Based Inference (SBI). In hierarchical settings with shared global parameters and exchangeable site-level parameters and observations, this structure can be exploited to improve simulation efficiency. Existing hierarchical SBI approaches factorise the posterior yet still simulate across multiple sites per training sample; We instead explore likelihood factorisation (LF) to train from single-site simulations. In LF sampling we learn a per-site neural surrogate of the simulator and then assemble synthetic multi-site observations to amortise inference for the full hierarchical posterior. Building on this, we propose Tokenised Flow Matching for Posterior Estimation (TFMPE), a tokenised flow matching approach that supports function-valued observations through likelihood factorisation. To enable systematic evaluation, we introduce a benchmark for hierarchical SBI. We validate TFMPE on this benchmark and on realistic infectious disease and computational fluid dynamics models, finding well-calibrated posteriors while reducing computational cost.

Digital twins of natural systems must remain aligned with physical systems that evolve over time, are only partially observed, and are typically modeled by mechanistic simulators whose parameters cannot be measured directly. In such settings, model adaptation is naturally posed as a simulation-based inference problem. However, sparse and indirect observations often fail to identify a unique and optimal calibration, leaving several simulator parameterizations compatible with the available evidence. This article presents a GFlowNet-based approach to model adaptation for digital twins of natural systems. We formulate adaptation as a generative modeling problem over complete simulator configurations, so that plausible parameterizations can be sampled with probability proportional to a reward derived from agreement between simulated and observed behavior. Using a controlled environment agriculture case study based on a mechanistic tomato model, we show that the learned policy recovers dominant regions of the adaptation landscape, retrieves strong calibration hypotheses, and preserves multiple plausible configurations under uncertainty.

Selection bias arises when the probability that an observation enters a dataset depends on variables related to the quantities of interest, leading to systematic distortions in estimation and uncertainty quantification. For example, in epidemiological or survey settings, individuals with certain outcomes may be more likely to be included, resulting in biased prevalence estimates with potentially substantial downstream impact. Classical corrections, such as inverse-probability weighting or explicit likelihood-based models of the selection process, rely on tractable likelihoods, which limits their applicability in complex stochastic models with latent dynamics or high-dimensional structure. Simulation-based inference enables Bayesian analysis without tractable likelihoods but typically assumes missingness at random and thus fails when selection depends on unobserved outcomes or covariates. Here, we develop a bias-aware simulation-based inference framework that explicitly incorporates selection into neural posterior estimation. By embedding the selection mechanism directly into the generative simulator, the approach enables amortized Bayesian inference without requiring tractable likelihoods. This recasting of selection bias as part of the simulation process allows us to both obtain debiased estimates and explicitly test for the presence of bias. The framework integrates diagnostics to detect discrepancies between simulated and observed data and to assess posterior calibration. The method recovers well-calibrated posterior distributions across three statistical applications with diverse selection mechanisms, including settings in which likelihood-based approaches yield biased estimates. These results recast the correction of selection bias as a simulation problem and establish simulation-based inference as a practical and testable strategy for parameter estimation under selection bias.

Diffusion models have recently emerged as powerful learners for simulation-based inference (SBI), enabling fast and accurate estimation of latent parameters from simulated and real data. Their score-based formulation offers a flexible way to learn conditional or joint distributions over parameters and observations, thereby providing a versatile solution to various modeling problems. In this tutorial review, we synthesize recent developments on diffusion models for SBI, covering design choices for training, inference, and evaluation. We highlight opportunities created by various concepts such as guidance, score composition, flow matching, consistency models, and joint modeling. Furthermore, we discuss how efficiency and statistical accuracy are affected by noise schedules, parameterizations, and samplers. Finally, we illustrate these concepts with case studies across parameter dimensionalities, simulation budgets, and model types, and outline open questions for future research.

Likelihood-free inference (LFI) enables system identification in complex tasks via black-box modelling, abstracting nonlinearity and stochasticity, and infers a domain distribution for adapting agents to parametric deployment conditions. LFI assumes an arbitrary support for sampling, which remains fixed as the initial generic prior is refined to increasingly descriptive posteriors. Misspecified support can therefore yield suboptimal yet overconfident posteriors. We address this issue by using the posterior of an inference step to guide the adaptation of the support using three illustrative heuristics: EDGE, MODE, and CENTRE. Each heuristic interprets the updated belief and enables support adaptation alongside posterior inference. For illustrative purposes, we first study misspecified support in LFI and evaluate the utility of our heuristics using stochastic dynamical benchmarks. We then evaluate posterior-driven heuristic support adaptation for parameter inference and policy learning in a dynamic deformable linear object (DLO) manipulation task. Inference results in a finer length and stiffness classification for a parametric set of DLOs. When the resulting posteriors are used as domain distributions for sim-based policy learning, they lead to more robust object-centric agent performance.

In this work, we propose a flow-matching Markov chain Monte Carlo (FM-MCMC) algorithm for estimating the orbital parameters of exoplanetary systems, especially for those only one exoplanet is involved. Compared to traditional methods that rely on random sampling within the Bayesian framework, our approach first leverages flow matching posterior estimation (FMPE) to efficiently constrain the prior range of physical parameters, and then employs MCMC to accurately infer the posterior distribution. For example, in the orbital parameter inference of beta Pictoris b, our model achieved a substantial speed-up while maintaining comparable accuracy-running 77.8 times faster than Parallel Tempered MCMC (PTMCMC) and 365.4 times faster than nested sampling. Moreover, our FM-MCMC method also attained the highest average log-likelihood among all approaches, demonstrating its superior sampling efficiency and accuracy. This highlights the scalability and efficiency of our approach, making it well-suited for processing the massive datasets expected from future exoplanet surveys. Beyond astrophysics, our methodology establishes a versatile paradigm for synergizing deep generative models with traditional sampling, which can be adopted to tackle complex inference problems in other fields, such as cosmology, biomedical imaging, and particle physics.

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.

We present a bifidelity method for uncertainty quantification of parameter estimates in complex systems, leveraging generative models trained to sample the target conditional distribution. In the Bayesian inference setting, traditional parameter estimation methods rely on repeated simulations of potentially expensive forward models to determine the posterior distribution of the parameter values, which may result in computationally intractable workflows. Furthermore, methods such as Markov Chain Monte Carlo (MCMC) necessitate rerunning the entire algorithm for each new data observation, further increasing the computational burden. Hence, we propose a novel method for efficiently obtaining posterior distributions of parameter estimates for high-fidelity models given data observations of interest. The method first constructs a low-fidelity, conditional generative model capable of amortized Bayesian inference and hence rapid posterior density approximation over a wide-range of data observations. When higher accuracy is needed for a specific data observation, the method employs adaptive refinement of the density approximation. It uses outputs from the low-fidelity generative model to refine the parameter sampling space, ensuring efficient use of the computationally expensive high-fidelity solver. Subsequently, a high-fidelity, unconditional generative model is trained to achieve greater accuracy in the target posterior distribution. Both low- and high- fidelity generative models enable efficient sampling from the target posterior and do not require repeated simulation of the high-fidelity forward model. We demonstrate the effectiveness of the proposed method on several numerical examples, including cases with multi-modal densities, as well as an application in plasma physics for a runaway electron simulation model.