arXiv subjects

Paul-Christian Bürkner

Publications and source records attributed to Paul-Christian Bürkner.

At least 19 recordsLinked to original sources

arXiv2026

Semi-Amortized Bayesian Inference with Reusable Local Posteriors

Amortized Bayesian inference uses simulations to perform Bayesian estimation when likelihoods are expensive or analytically unavailable. However, learning the full joint posterior is challenging in high-dimensional models with complex dependencies. We introduce a semi-amortized method that divides parameters into smaller blocks, and uses conditional independence to identify the information needed to infer each block. We train neural estimators for blocks whose posterior updates are computationally expensive or frequently repeated. Each estimator learns from simulations of the relevant model components to approximate the block's full conditional posterior, given the relevant observations and the values of the variables on which it depends. A Markov chain then alternates direct draws from these estimators with conventional updates of the remaining variables. Each neural update uses the current conditioning values and requires neither likelihood evaluation nor an acceptance-rejection step. Where the same conditional structure recurs, one estimator can be reused across blocks and datasets, with an output dimension determined by the block rather than the full model. We illustrate the approach in high-dimensional and computationally expensive hierarchical and spatiotemporal models.

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arXiv2026

Amortized Bayesian Inference on Multilevel Models of Arbitrary Structure

We develop a general method for amortized Bayesian inference on multilevel models of arbitrary structure. Given a generative model specified as a directed acyclic graph, our method automatically derives valid factorizations of the joint posterior and matching neural network architectures. The key steps, graph expansion and graph inversion, yield an inverse graph that determines how inference networks are stacked and conditioned, producing factorizations that amortize over the number of groups and the number of observations within each group. Unlike approaches that simplify the dependency structure to speed up learning or inference, our method preserves all conditional independence and exchangeability assumptions of the generative model. Across three case studies, it closely matches gold-standard samplers on models with more than 6,500 parameters while reducing inference to a near-instant forward pass once trained.

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arXiv2026

High-Dimensional Simulation-Based Inference in Latent Spaces

Neural simulation-based inference (SBI) has been widely successful in inferring a relatively small number of interpretable parameters from potentially high-dimensional observations, such as images or time series. Accordingly, representation learning in SBI has focused almost exclusively on compressing the observations used to condition the posterior. More recently, however, SBI has begun to target increasingly high-dimensional parameter spaces, raising the complementary question of whether the inference target itself should be compressed. Our answer is a practical merger of SBI and latent generative modeling, which learns a low-dimensional representation of the simulator parameters, performs posterior inference directly in this latent space, and maps posterior samples back to the original parameter space. We characterize the conditions under which latent-space inference recovers the desired target posterior and systematically study its empirical trade-offs. Across four case studies and three generative families, we compare latent and standard estimators while controlling for network capacity, regularization, optimization, and training compute. At matched training compute, latent-space inference achieves accuracy and marginal calibration comparable to direct target-space inference while sampling up to more than an order of magnitude faster.

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arXiv2026

Approximating Bayesian leave-one-group-out cross-validation

When data are grouped, hierarchical or multilevel models are commonly used to account for group-level variation with group-specific parameters. Leave-one-group-out cross-validation (LOGO-CV) is a suitable tool for evaluating predictive performance for new groups, providing an estimator of the expected log predictive density (elpd). Brute-force LOGO-CV requires one model refit per held-out group, often using computationally expensive inference algorithms such as MCMC. This is costly, particularly for large numbers of groups or complex model structures. Commonly used importance sampling approximations, intended to reduce this cost, tend to fail because the group-specific parameters of the held-out group must be integrated out. We identify two key challenges in LOGO-CV elpd estimation: approximating the LOGO posterior and computing the grouped marginal likelihood. We compare 11 strategies, including 5 newly proposed, to address them. Among others, we combine Pareto-smoothed importance sampling or adaptive importance sampling with integration techniques such as Laplace approximation, adaptive Gauss-Hermite quadrature, and bridge sampling. We evaluate these strategies in both simulation experiments and real-world case studies, which show that marginalising over the group-specific parameters substantially improves the reliability of the importance sampling approaches.

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arXiv2026

Efficient Uncertainty Propagation in Bayesian Two-Step Procedures

Bayesian inference provides a principled framework for probabilistic reasoning. If inference is performed in two steps, uncertainty propagation plays a crucial role in accounting for all sources of uncertainty and variability. This becomes particularly important when both aleatoric uncertainty, caused by data variability, and epistemic uncertainty, arising from incomplete knowledge or missing data, are present. Examples include surrogate models and missing data problems. In surrogate modeling, the surrogate is used as a simplified approximation of a resource-heavy and costly simulation. The uncertainty from the surrogate-fitting process can be propagated using a two-step procedure. For modeling with missing data, methods like Multivariate Imputation by Chained Equations (MICE) generate multiple datasets to account for imputation uncertainty. These approaches, however, are computationally expensive, as multiple models must be fitted separately to surrogate parameters respectively imputed datasets. To address these challenges, we propose an efficient two-step approach that reduces computational overhead while maintaining accuracy. By selecting a representative subset of draws or imputations, we construct a mixture distribution to approximate the desired posteriors using Pareto smoothed importance sampling. For more complex scenarios, this is further refined with importance weighted moment matching and an iterative procedure that broadens the mixture distribution to better capture diverse posterior distributions.

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arXiv2026

Simulation-based validation of Bayes factor computation

We propose and evaluate two methods that validate the computation of Bayes factors: one based on an improved variant of simulation-based calibration checking (SBC) and one based on calibration metrics for binary predictions. We show that in theory, binary prediction calibration is equivalent to a special case of SBC, but with limited resources, binary prediction calibration is typically more sensitive to the problems we investigated. With well-designed test quantities, SBC can however detect all possible problems in computation, including some that cannot be uncovered by binary prediction calibration. Previous work on Bayes factor validation includes checks based on the data-averaged posterior and the Good check method. We demonstrate that both checks miss many problems in Bayes factor computation detectable with SBC and binary prediction calibration. Moreover, we find that the Good check as originally described fails to control its error rates. Our proposed checks also typically use simulation results more efficiently than data-averaged posterior checks. Finally, we show that a special approach based on posterior SBC is necessary when checking Bayes factor computation under improper priors and we validate several models with such priors. We recommend that novel methods for Bayes factor computation be validated with SBC, binary prediction calibration and data-averaged posterior with at least several hundred simulations. For all the models we tested, the bridgesampling and BayesFactor R packages satisfy all available checks and thus are likely safe to use in standard scenarios.

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arXiv2026

JADAI: Jointly Amortizing Adaptive Design and Bayesian Inference

We consider problems of parameter estimation where design variables can be actively optimized to maximize information gain. To this end, we introduce JADAI, a framework that jointly amortizes Bayesian adaptive design and inference by training a policy, a history network, and an inference network end-to-end. The networks minimize a generic loss that aggregates incremental reductions in posterior error along experimental sequences. Inference networks are instantiated with diffusion-based posterior estimators that can approximate high-dimensional and multimodal posteriors at every experimental step. Across standard adaptive design benchmarks, JADAI achieves superior or competitive performance.

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arXiv2026

To select or not to select: predictively consistent priors instead of model selection

Bayesian modelling workflows often consider multiple candidate models of varying complexity. Model selection is commonly used to navigate potential trade-offs between model complexity and generalisability to new data. We study when model selection is unnecessary or can even be harmful for predictive performance in finite data regimes and find that the need for selecting simpler models can depend on prior choice. We formalise predictively consistent priors, which keep prior predictive implications stable as model complexity increases. Across examples and numerical experiments, including adding covariates in linear and logistic regression, forward variable selection, and nonlinear modelling, flexible models with predictively consistent priors typically match or outperform selected simpler models in out-of-sample predictive performance. When selection helps, it can indicate poor joint prior implications, such as excessive prior mass on implausible predictive values. Based on our findings, we propose replacing the notion of sparsity or parsimony at the level of model components with specifying priors that remain sensible in predictive space as models become more complex.

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arXiv2026

Latent Variable Models for Distributional Features

Analyzing the mean response of study subjects in psychological research is a standard, well-justified practice. However, theoretical arguments and empirical evidence also suggest that there is value in investigating other aspects of the distribution of such responses, such as their variability or skewness. A particular challenge that practitioners face is statistical modeling of associations between distributional features and other outcomes of interest. The most common approach is to perform estimation in two steps: distributional features are estimated first, and then those estimates are used as predictors for the relevant outcomes. Such an approach is most amenable to implementation in standard statistical software, but it ignores estimation error and can therefore lead to biased estimates and increased error rates. We introduce Distributional Feature Latent Variable Models (DFLVM), a general framework that represents between-person difference in distributional features as random intercepts. These intercepts can be simultaneously used as predictors for downstream outcomes and their associations estimated in a single estimation step. We compare the performance of our approach against two-step procedures in a simulation study and through a re-analysis of a real dataset.

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arXiv2026

Bayesian Surrogate Training on Multiple Data Sources: A Hybrid Modeling Strategy

Surrogate models are often used as computationally efficient approximations to complex simulation models, enabling tasks such as solving inverse problems, sensitivity analysis, and probabilistic forward predictions, which would otherwise be computationally infeasible. During training, surrogate parameters are fitted such that the surrogate reproduces the simulation model's outputs as closely as possible. However, the simulation model itself is merely a simplification of the real-world system, often missing relevant processes or suffering from misspecifications e.g., in inputs or boundary conditions. Hints about these might be captured in real-world measurement data, and yet, we typically ignore those hints during surrogate building. In this paper, we propose two novel probabilistic approaches to integrate simulation data and real-world measurement data during surrogate training. The first method trains separate surrogate models for each data source and combines their predictive distributions, while the second incorporates both data sources by training a single surrogate. Both hybrid modeling approaches employ a novel weighting strategy for combining heterogeneous data sources during surrogate training, which operates independently of the chosen surrogate family. We show the conceptual differences and benefits of the two approaches through both synthetic and real-world case studies. The results demonstrate the potential of these methods to improve predictive accuracy, predictive coverage, and to diagnose problems in the underlying simulation model. These insights can improve system understanding and future model development.

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arXiv2026

Improving the Accuracy of Amortized Model Comparison with Self-Consistency

Amortized Bayesian model comparison (BMC) enables fast probabilistic ranking of models via simulation-based training of neural surrogates. However, the accuracy of neural surrogates deteriorates when simulation models are misspecified; the very case where model comparison is most needed. We evaluate four different amortized BMC methods. We supplement traditional simulation-based training of these methods with a \emph{self-consistency} (SC) loss on unlabeled real data to improve BMC estimates under distribution shifts. Using one artificial and two real-world case studies, we compare amortized BMC estimators with and without SC against analytic or bridge sampling benchmarks. In the \emph{closed-world} case (data is generated by one of the candidate models), BMC estimators using classifiers work acceptably well even without SC training. However, these methods also benefit the least from SC training. In the \emph{open-world} scenario (all models misspecified), SC training strongly improves BMC estimators when having access to analytic likelihoods, or when surrogate likelihoods are locally accurate near the true parameter posterior, even for severely misspecified models. We conclude with practical recommendations for amortized BMC and suggestions for future research.

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arXiv2026

From Mice to Trains: Amortized Bayesian Inference on Graph Data

Graphs arise across diverse domains, from biology and chemistry to social and information networks, as well as in transportation and logistics. Inference on graph-structured data requires methods that are permutation-invariant, scalable across varying sizes and sparsities, and capable of capturing complex long-range dependencies, making posterior estimation on graph parameters particularly challenging. Amortized Bayesian Inference (ABI) is a simulation-based framework that employs generative neural networks to enable fast, likelihood-free posterior inference. We adapt ABI to graph data to address these challenges to perform inference on node-, edge-, and graph-level parameters. Our approach couples permutation-invariant graph encoders with flexible neural posterior estimators in a two-module pipeline: a summary network maps attributed graphs to fixed-length representations, and an inference network approximates the posterior over parameters. In this setting, several neural architectures can serve as the summary network. In this work we evaluate multiple architectures and assess their performance on controlled synthetic settings and two real-world domains - biology and logistics - in terms of recovery and calibration.

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arXiv2026

BayesFlow 2: Multi-Backend Amortized Bayesian Inference in Python

Modern Bayesian inference involves a mixture of computational methods for estimating, validating, and drawing conclusions from probabilistic models as part of principled workflows. An overarching motif of many Bayesian methods is that they are relatively slow, which often becomes prohibitive when fitting complex models to large data sets. Amortized Bayesian inference (ABI) offers a path to solving the computational challenges of Bayes. ABI trains neural networks on model simulations, rewarding users with rapid inference of any model-implied quantity, such as point estimates, likelihoods, or full posterior distributions. In this work, we present the Python library BayesFlow, Version 2.0, for general-purpose ABI. Along with direct posterior, likelihood, and ratio estimation, the software includes support for multiple popular deep learning backends, a rich collection of generative networks for sampling and density estimation, complete customization and high-level interfaces, as well as new capabilities for hyperparameter optimization, design optimization, and hierarchical modeling. Using a case study on dynamical system parameter estimation, combined with comparisons to similar software, we show that our streamlined, user-friendly workflow has strong potential to support broad adoption.

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arXiv2026

Robust Amortized Bayesian Inference with Self-Consistency Losses on Unlabeled Data

Amortized Bayesian inference (ABI) with neural networks can solve probabilistic inverse problems orders of magnitude faster than classical methods. However, ABI is not yet sufficiently robust for widespread and safe application. When performing inference on observations outside the scope of the simulated training data, posterior approximations are likely to become highly biased, which cannot be corrected by additional simulations due to the bad pre-asymptotic behavior of current neural posterior estimators. In this paper, we propose a semi-supervised approach that enables training not only on labeled simulated data generated from the model, but also on \textit{unlabeled} data originating from any source, including real data. To achieve this, we leverage Bayesian self-consistency properties that can be transformed into strictly proper losses that do not require knowledge of ground-truth parameters. We test our approach on several real-world case studies, including applications to high-dimensional time-series and image data. Our results show that semi-supervised learning with unlabeled data drastically improves the robustness of ABI in the out-of-simulation regime. Notably, inference remains accurate even when evaluated on observations far away from the labeled and unlabeled data seen during training.

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arXiv2026

Unsupervised Continual Learning for Amortized Bayesian Inference

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.

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arXiv2026

Amortized Bayesian Workflow

Bayesian inference often faces a trade-off between computational speed and sampling accuracy. We propose an adaptive workflow that integrates rapid amortized inference with gold-standard MCMC techniques to achieve a favorable combination of both speed and accuracy when performing inference on many observed datasets. Our approach uses principled diagnostics to guide the choice of inference method for each dataset, moving along the Pareto front from fast amortized sampling via generative neural networks to slower but guaranteed-accurate MCMC when needed. By reusing computations across steps, our workflow synergizes amortized and MCMC-based inference. We demonstrate the effectiveness of this integrated approach on several synthetic and real-world problems with tens of thousands of datasets, showing efficiency gains while maintaining high posterior quality.

cs.LG↗

arXiv2026

Hilbert space methods for approximating multi-output latent variable Gaussian processes

Gaussian processes are a powerful class of non-linear models, but have limited applicability for larger datasets due to their high computational complexity. In such cases, approximate methods are required, for example, the recently developed class of Hilbert space Gaussian processes. They have been shown to significantly reduce computation time while retaining most of the favorable properties of exact Gaussian processes. However, Hilbert space approximations have so far only been developed for uni-dimensional outputs and manifest (known) inputs. Thus, we generalize Hilbert space methods to multi-output and latent input settings. Through extensive simulations, we show that the developed approximate Gaussian processes are indeed not only faster, but also provide similar or even better uncertainty calibration and accuracy of latent variable estimates compared to exact Gaussian processes. While not necessarily faster than alternative Gaussian process approximations, our new models provide better calibration and estimation accuracy, thus striking an excellent balance between trustworthiness and speed. We additionally illustrate our methods on a real-world case study from single cell biology.

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arXiv2026

Improving the Accuracy of Amortized Model Comparison with Self-Consistency

Amortized Bayesian inference (ABI) offers fast, scalable approximations to posterior densities by training neural surrogates on data simulated from the statistical model. However, ABI methods are highly sensitive to model misspecification: when observed data fall outside the training distribution (generative scope of the statistical models), neural surrogates can behave unpredictably. This makes it a challenge in a model comparison setting, where multiple statistical models are considered, of which at least some are misspecified. Recent work on self-consistency (SC) provides a promising remedy to this issue, accessible even for empirical data (without ground-truth labels). In this work, we investigate how SC can improve amortized model comparison conceptualized in four different ways. Across two synthetic and two real-world case studies, we find that approaches for model comparison that estimate marginal likelihoods through approximate parameter posteriors consistently outperform methods that directly approximate model evidence or posterior model probabilities. SC training improves robustness when the likelihood is available, even under severe model misspecification. The benefits of SC for methods without access of analytic likelihoods are more limited and inconsistent. Our results suggest practical guidance for reliable amortized Bayesian model comparison: prefer parameter posterior-based methods and augment them with SC training on empirical datasets to mitigate extrapolation bias under model misspecification.

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