This textbook gives a systematic treatment of statistical machine learning for astronomical research, based on Bayesian inference. It shows how modern data analysis techniques connect to classical statistical methods and follow from familiar statistical principles. The Bayesian perspective, used throughout, gives priority to uncertainty quantification and statistical rigor, which scientific inference in astronomy needs. The textbook progresses from probability theory and Bayesian inference through supervised learning, including linear regression with measurement uncertainties, logistic regression and classification. Unsupervised learning topics cover Principal Component Analysis and clustering methods. We then introduce sampling and Markov Chain Monte Carlo, followed by Gaussian Processes as probabilistic nonparametric methods and neural networks in the broader statistical context. Our theory-focused approach derives each method from first principles with complete mathematical development and then shows its astronomical applications. We put statistical insight first and explain why algorithms work, when they are appropriate, and how they connect to broader statistical principles. From a solid grounding in classical methods and their theory, the treatment leads to modern techniques such as neural networks. With this grounding, readers can apply the methods to astronomical research with proper attention to assumptions, limits and uncertainty propagation, as large astronomical surveys require. The 22 companion tutorials apply the methods to astronomical data in Jupyter notebooks.
Stochastic Petri Nets (SPNs) are an increasingly popular tool of choice for modeling discrete-event dynamics in areas such as epidemiology and systems biology, yet their parameter estimation remains challenging in general and in particular when transition rates depend on external covariates and explicit likelihoods are unavailable. We introduce a neural-surrogate (neural-network--based approximation of the posterior distribution) framework that predicts the coefficients of known covariate-dependent rate functions directly from noisy, partially observed token trajectories. Our model employs a lightweight 1D Convolutional Residual Network trained end-to-end on Gillespie-simulated SPN realizations, learning to invert system dynamics under realistic conditions of event dropout. During inference, Monte Carlo dropout provides calibrated uncertainty bounds together with point estimates. On synthetic SPNs with 20% missing events, our surrogate recovers rate-function coefficients with an RMSE = 0.108 and substantially runs faster than traditional Bayesian approaches. These results demonstrate that data-driven, likelihood-free surrogates can enable accurate, robust, and real-time parameter recovery in complex, partially observed discrete-event systems.
Despite their widespread use, large language models (LLMs) are known to hallucinate incorrect information and be poorly calibrated. This makes the uncertainty quantification of these models of critical importance, especially in high-stakes domains, such as autonomy and healthcare. Prior work has made Bayesian deep learning-based approaches to this problem more tractable by performing inference over the low-rank adaptation (LoRA) parameters of a fine-tuned model. While effective, these approaches struggle to scale to larger LLMs due to requiring further additional parameters compared to LoRA. In this work we present $\textbf{Scala}$ble $\textbf{B}$ayesian $\textbf{L}$ow-Rank Adaptation via Stochastic Variational Subspace Inference (ScalaBL). We perform Bayesian inference in an $r$-dimensional subspace, for LoRA rank $r$. By repurposing the LoRA parameters as projection matrices, we are able to map samples from this subspace into the full weight space of the LLM. This allows us to learn all the parameters of our approach using stochastic variational inference. Despite the low dimensionality of our subspace, we are able to achieve competitive performance with state-of-the-art approaches while only requiring ${\sim}1000$ additional parameters. Furthermore, it allows us to scale up to the largest Bayesian LLM to date, with four times as a many base parameters as prior work.
This paper focuses on wireless multiple-input multiple-output (MIMO)-orthogonal frequency division multiplex (OFDM) receivers. Traditional wireless receivers have relied on mathematical modeling and Bayesian inference, achieving remarkable success in most areas but falling short in their ability to characterize channel matrices. Neural networks (NNs) have demonstrated significant potential in this aspect. Nevertheless, integrating traditional inference methods with NNs presents challenges, particularly in tracking the error progression. Given the inevitable presence of noise in wireless systems, generative models that are more resilient to noise are garnering increased attention. In this paper, we propose re-evaluating the MIMO-OFDM receiver using diffusion models, which is a common generative approach. With diffusion models, we can effectively leverage prior knowledge of channel matrices and incorporate traditional signal estimation components. Specifically, we explore the diffusion system and introduce an imagination-screening strategy to guide the diffusion process. Furthermore, diffusion models enable adaptation to varying noise levels and pilot schemes using the same NN, significantly reducing training and deployment costs. Simulated results reveal that, for pilot densities ranging from 4-6 pilots per 64-subcarrier block and signal-to-noise ratios (SNRs) from -4 dB to 0 dB, our proposed receiver reduces channel-reconstruction error by up to two times compared to leading deep-learning models, with the most pronounced improvements observed in low-pilot conditions. Additionally, performance enhancements can be achieved with a larger imagination size, despite increased computational complexity.
Variational inference (VI) is a popular approach in Bayesian inference, that looks for the best approximation of the posterior distribution within a parametric family, minimizing a loss that is typically the (reverse) Kullback-Leibler (KL) divergence. In this paper, we focus on the following parametric family: mixtures of isotropic Gaussians (i.e., with diagonal covariance matrices proportional to the identity) and uniform weights. We develop a variational framework and provide efficient algorithms suited for this family. In contrast with mixtures of Gaussian with generic covariance matrices, this choice presents a balance between accurate approximations of multimodal Bayesian posteriors, while being memory and computationally efficient. Our algorithms implement gradient descent on the location of the mixture components (the modes of the Gaussians), and either (an entropic) Mirror or Bures descent on their variance parameters. We illustrate the performance of our algorithms on numerical experiments.
Neural simulation-based inference (SBI) is a popular set of methods for Bayesian inference when models are only available in the form of a simulator. These methods are widely used in the sciences and engineering, where writing down a likelihood can be significantly more challenging than constructing a simulator. However, the performance of neural SBI can suffer when simulators are computationally expensive, thereby limiting the number of simulations that can be performed. In this paper, we propose a novel approach to neural SBI which leverages multilevel Monte Carlo techniques for settings where several simulators of varying cost and fidelity are available. We demonstrate through both theoretical analysis and extensive experiments that our method can significantly enhance the accuracy of SBI methods given a fixed computational budget.
Many critical applications, from autonomous scientific discovery to personalized medicine, demand systems that can both strategically acquire the most informative data and instantaneously perform inference based upon it. While amortized methods for Bayesian inference and experimental design offer part of the solution, neither approach is optimal in the most general and challenging task, where new data needs to be collected for instant inference. To tackle this issue, we introduce the Amortized Active Learning and Inference Engine (ALINE), a unified framework for amortized Bayesian inference and active data acquisition. ALINE leverages a transformer architecture trained via reinforcement learning with a reward based on self-estimated information gain provided by its own integrated inference component. This allows it to strategically query informative data points while simultaneously refining its predictions. Moreover, ALINE can selectively direct its querying strategy towards specific subsets of model parameters or designated predictive tasks, optimizing for posterior estimation, data prediction, or a mixture thereof. Empirical results on regression-based active learning, classical Bayesian experimental design benchmarks, and a psychometric model with selectively targeted parameters demonstrate that ALINE delivers both instant and accurate inference along with efficient selection of informative points.
The expressiveness of flow-based models combined with stochastic variational inference (SVI) has, in recent years, expanded the application of optimization-based Bayesian inference to include problems with complex data relationships. However, until now, SVI using flow-based models has been limited to problems of fixed dimension. We introduce CoSMIC, normalizing flows (COntextually-Specified Masking for Identity-mapped Components), an extension to neural autoregressive conditional normalizing flow architectures that enables using a single amortized variational density for inference over a transdimensional target distribution. We propose a combined stochastic variational transdimensional inference (VTI) approach to training CoSMIC flows using techniques from Bayesian optimization and Monte Carlo gradient estimation. Numerical experiments demonstrate the performance of VTI on challenging problems that scale to high-cardinality model spaces.
Hollmann et al. (Nature 637 (2025) 319-326) recently introduced TabPFN, a transformer-based deep learning model for regression and classification on tabular data, which they claim "outperforms all previous methods on datasets with up to 10,000 samples by a wide margin, using substantially less training time." Furthermore, they have called TabPFN a "foundation model" for tabular data, as it can support "data generation, density estimation, learning reusable embeddings and fine-tuning". If these statements are well-supported, TabPFN may have the potential to supersede existing modeling approaches on a wide range of statistical tasks, mirroring a similar revolution in other areas of artificial intelligence that began with the advent of large language models. In this paper, we provide a tailored explanation of how TabPFN works for a statistics audience, by emphasizing its interpretation as approximate Bayesian inference. We also provide more evidence of TabPFN's "foundation model" capabilities: We show that an out-of-the-box application of TabPFN vastly outperforms specialized state-of-the-art methods for semi-supervised parameter estimation, prediction under covariate shift, and heterogeneous treatment effect estimation. We further show that TabPFN can outperform LASSO at sparse regression and can break a robustness-efficiency trade-off in classification. All experiments can be reproduced using the code provided at https://github.com/qinglong-tian/tabpfn_study (https://github.com/qinglong-tian/tabpfn_study).
Adapting to the addressee is crucial for successful explanations, yet poses significant challenges for dialogsystems. We adopt the approach of treating explanation generation as a non-stationary decision process, where the optimal strategy varies according to changing beliefs about the explainee and the interaction context. In this paper we address the questions of (1) how to track the interaction context and the relevant listener features in a formally defined computational partner model, and (2) how to utilize this model in the dynamically adjusted, rational decision process that determines the currently best explanation strategy. We propose a Bayesian inference-based approach to continuously update the partner model based on user feedback, and a non-stationary Markov Decision Process to adjust decision-making based on the partner model values. We evaluate an implementation of this framework with five simulated interlocutors, demonstrating its effectiveness in adapting to different partners with constant and even changing feedback behavior. The results show high adaptivity with distinct explanation strategies emerging for different partners, highlighting the potential of our approach to improve explainable AI systems and dialogsystems in general.
Network interference occurs when treatments assigned to some units affect the outcomes of others. Traditional approaches often assume that the observed network correctly specifies the interference structure. However, in practice, researchers frequently only have access to proxy measurements of the interference network due to limitations in data collection or potential mismatches between measured networks and actual interference pathways. In this paper, we introduce a framework for estimating causal effects when only proxy networks are available. Our approach leverages a structural causal model that accommodates diverse proxy types, including noisy measurements, multiple data sources, and multilayer networks, and defines causal effects as interventions on population-level treatments. Since the true interference network is latent, estimation poses significant challenges. To overcome them, we develop a Bayesian inference framework. We propose a Block Gibbs sampler with Locally Informed Proposals to update the latent network, thereby efficiently exploring the high-dimensional posterior space composed of both discrete and continuous parameters. We illustrate the performance of our method through numerical experiments, demonstrating its accuracy in recovering causal effects even when only proxies of the interference network are available.
Federated Learning enables collaborative training of machine learning models on decentralized data. This scheme, however, is vulnerable to adversarial attacks, when some of the clients submit corrupted model updates. In real-world scenarios, the total number of compromised clients is typically unknown, with the extent of attacks potentially varying over time. To address these challenges, we propose an adaptive approach for robust aggregation of model updates based on Bayesian inference. The mean update is defined by the maximum of the likelihood marginalized over probabilities of each client to be `honest'. As a result, the method shares the simplicity of the classical average estimators (e.g., sample mean or geometric median), being independent of the number of compromised clients. At the same time, it is as effective against attacks as methods specifically tailored to Federated Learning, such as Krum. We compare our approach with other aggregation schemes in federated setting on three benchmark image classification data sets. The proposed method consistently achieves state-of-the-art performance across various attack types with static and varying number of malicious clients.
Studies of hadron resonances and their properties are limited by the accuracy and consistency of measured datasets, which can originate from many different experiments. We have used Gaussian Processes (GP) to build interpolated datasets, including quantification of uncertainties, so that data from different sources can be used in model fitting without the need for arbitrary weighting. GPs predict values and uncertainties of observables at any kinematic point. Bayesian inference is used to optimise the hyperparameters of the GP model. We demonstrate that the GP successfully interpolates data with quantified uncertainties by comparison with generated pseudodata. We also show that this methodology can be used to investigate the consistency of data from different sources. GPs provide a robust, model-independent method for interpolating typical datasets used in hadron resonance studies, removing the limitations of arbitrary weighting in sparse datasets.
Magnetic resonance spectroscopy (MRS) is a non-invasive technique to measure the metabolic composition of tissues, offering valuable insights into neurological disorders, tumor detection, and other metabolic dysfunctions. However, accurate metabolite quantification is hindered by challenges such as spectral overlap, low signal-to-noise ratio, and various artifacts. Traditional methods like linear-combination modeling are susceptible to ambiguities and commonly only provide a theoretical lower bound on estimation accuracy in the form of the Cram\'er-Rao bound. This work introduces a Bayesian inference framework using Sylvester normalizing flows (SNFs) to approximate posterior distributions over metabolite concentrations, enhancing quantification reliability. A physics-based decoder incorporates prior knowledge of MRS signal formation, ensuring realistic distribution representations. We validate the method on simulated 7T proton MRS data, demonstrating accurate metabolite quantification, well-calibrated uncertainties, and insights into parameter correlations and multi-modal distributions.
In semantic segmentation, the accuracy of models heavily depends on the high-quality annotations. However, in many practical scenarios such as medical imaging and remote sensing, obtaining true annotations is not straightforward and usually requires significant human labor. Relying on human labor often introduces annotation errors, including mislabeling, omissions, and inconsistency between annotators. In the case of remote sensing, differences in procurement time can lead to misaligned ground truth annotations. These label errors are not independently distributed, and instead usually appear in spatially connected regions where adjacent pixels are more likely to share the same errors. To address these issues, we propose an approximate Bayesian estimation based on a probabilistic model that assumes training data includes label errors, incorporating the tendency for these errors to occur with spatial correlations between adjacent pixels. Bayesian inference requires computing the posterior distribution of label errors, which becomes intractable when spatial correlations are present. We represent the correlation of label errors between adjacent pixels through a Gaussian distribution whose covariance is structured by a Kac-Murdock-Szeg\"{o} (KMS) matrix, solving the computational challenges. Through experiments on multiple segmentation tasks, we confirm that leveraging the spatial correlation of label errors significantly improves performance. Notably, in specific tasks such as lung segmentation, the proposed method achieves performance comparable to training with clean labels under moderate noise levels. Code is available at https://github.com/pfnet-research/Bayesian_SpatialCorr.
Biological synapses effortlessly balance memory retention and flexibility, yet artificial neural networks still struggle with the extremes of catastrophic forgetting and catastrophic remembering. Here, we introduce Metaplasticity from Synaptic Uncertainty (MESU), a Bayesian framework that updates network parameters according their uncertainty. This approach allows a principled combination of learning and forgetting that ensures that critical knowledge is preserved while unused or outdated information is gradually released. Unlike standard Bayesian approaches -- which risk becoming overly constrained, and popular continual-learning methods that rely on explicit task boundaries, MESU seamlessly adapts to streaming data. It further provides reliable epistemic uncertainty estimates, allowing out-of-distribution detection, the only computational cost being to sample the weights multiple times to provide proper output statistics. Experiments on image-classification benchmarks demonstrate that MESU mitigates catastrophic forgetting, while maintaining plasticity for new tasks. When training 200 sequential permuted MNIST tasks, MESU outperforms established continual learning techniques in terms of accuracy, capability to learn additional tasks, and out-of-distribution data detection. Additionally, due to its non-reliance on task boundaries, MESU outperforms conventional learning techniques on the incremental training of CIFAR-100 tasks consistently in a wide range of scenarios. Our results unify ideas from metaplasticity, Bayesian inference, and Hessian-based regularization, offering a biologically-inspired pathway to robust, perpetual learning.
We propose a robust and scalable framework for variational Bayes (VB) that effectively handles outliers and contamination of arbitrary nature in large datasets. Our approach divides the dataset into disjoint subsets, computes the posterior for each subset, and applies VB approximation independently to these posteriors. The resulting variational posteriors with respect to the subsets are then aggregated using the geometric median of probability measures, computed with respect to the Wasserstein distance. This novel aggregation method yields the Variational Median Posterior (VM-Posterior) distribution. We rigorously demonstrate that the VM-Posterior preserves contraction properties akin to those of the true posterior, while accounting for approximation errors or the variational gap inherent in VB methods. We also provide provable robustness guarantee of the VM-Posterior. Furthermore, we establish a variational Bernstein-von Mises theorem for both multivariate Gaussian distributions with general covariance structures and the mean-field variational family. To facilitate practical implementation, we adapt existing algorithms for computing the VM-Posterior and evaluate its performance through extensive numerical experiments. The results highlight its robustness and scalability, making it a reliable tool for Bayesian inference in the presence of complex, contaminated datasets.
The analysis of data from multiple experiments, such as observations of several individuals, is commonly approached using mixed-effects models, which account for variation between individuals through hierarchical representations. This makes mixed-effects models widely applied in fields such as biology, pharmacokinetics, and sociology. In this work, we propose a novel methodology for scalable Bayesian inference in hierarchical mixed-effects models. Our framework first constructs amortized approximations of the likelihood and the posterior distribution, which are then rapidly refined for each individual dataset, to ultimately approximate the parameters posterior across many individuals. The framework is easily trainable, as it uses mixtures of experts but without neural networks, leading to parsimonious yet expressive surrogate models of the likelihood and the posterior. We demonstrate the effectiveness of our methodology using challenging stochastic models, such as mixed-effects stochastic differential equations emerging in systems biology-driven problems. However, the approach is broadly applicable and can accommodate both stochastic and deterministic models. We show that our approach can seamlessly handle inference for many parameters. Additionally, we applied our method to a real-data case study of mRNA transfection. When compared to exact pseudomarginal Bayesian inference, our approach proved to be both fast and competitive in terms of statistical accuracy.
We address the problem of uncertainty propagation in the discrete Fourier transform by modeling the fast Fourier transform as a factor graph. Building on this representation, we propose an efficient framework for approximate Bayesian inference using belief propagation (BP) and expectation propagation, extending its applicability beyond Gaussian assumptions. By leveraging an appropriate BP message representation and a suitable schedule, our method achieves stable convergence with accurate mean and variance estimates. Numerical experiments in representative scenarios from communications demonstrate the practical potential of the proposed framework for uncertainty-aware inference in probabilistic systems operating across both time and frequency domain.
Stein variational gradient descent (SVGD) is a kernel-based particle method for sampling from a target distribution, e.g., in generative modeling and Bayesian inference. SVGD does not require estimating the gradient of the log-density, which is called score estimation. In practice, SVGD can be slow compared to score-estimation based sampling algorithms. To design fast and efficient high-dimensional sampling algorithms, we introduce ASVGD, an accelerated SVGD, based on an accelerated gradient flow in a metric space of probability densities following Nesterov's method. We then derive a momentum-based discrete-time sampling algorithm, which evolves a set of particles deterministically. To stabilize the particles' momentum update, we also study a Wasserstein metric regularization. For the generalized bilinear kernel and the Gaussian kernel, toy numerical examples with varied target distributions demonstrate the effectiveness of ASVGD compared to SVGD and other popular sampling methods.