Parameter estimation is a fundamental challenge in machine learning, crucial for tasks such as neural network weight fitting and Bayesian inference. This paper focuses on the complexity of estimating translation $\boldsymbol{\mu} \in \mathbb{R}^l$ and shrinkage $\sigma \in \mathbb{R}_{++}$ parameters for a distribution of the form $\frac{1}{\sigma^l} f_0 \left( \frac{\boldsymbol{x} - \boldsymbol{\mu}}{\sigma} \right)$, where $f_0$ is a known density in $\mathbb{R}^l$ given $n$ samples. We highlight that while the problem is NP-hard for Maximum Likelihood Estimation (MLE), it is possible to obtain $\varepsilon$-approximations for arbitrary $\varepsilon > 0$ within $\text{poly} \left( \frac{1}{\varepsilon} \right)$ time using the Wasserstein distance.
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.
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.
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.
In scientific domains -- from biology to the social sciences -- many questions boil down to \textit{What effect will we observe if we intervene on a particular variable?} If the causal relationships (e.g.~a causal graph) are known, it is possible to estimate the intervention distributions. In the absence of this domain knowledge, the causal structure must be discovered from the available observational data. However, observational data are often compatible with multiple causal graphs, making methods that commit to a single structure prone to overconfidence. A principled way to manage this structural uncertainty is via Bayesian inference, which averages over a posterior distribution on possible causal structures and functional mechanisms. Unfortunately, the number of causal structures grows super-exponentially with the number of nodes in the graph, making computations intractable. We propose to circumvent these challenges by using meta-learning to create an end-to-end model: the Model-Averaged Causal Estimation Transformer Neural Process (MACE-TNP). The model is trained to predict the Bayesian model-averaged interventional posterior distribution, and its end-to-end nature bypasses the need for expensive calculations. Empirically, we demonstrate that MACE-TNP outperforms strong Bayesian baselines. Our work establishes meta-learning as a flexible and scalable paradigm for approximating complex Bayesian causal inference, that can be scaled to increasingly challenging settings in the future.
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.
Parameter estimation is a foundational step in statistical modeling, enabling us to extract knowledge from data and apply it effectively. Bayesian estimation of parameters incorporates prior beliefs with observed data to infer distribution parameters probabilistically and robustly. Moreover, it provides full posterior distributions, allowing uncertainty quantification and regularization, especially useful in small or truncated samples. Utilizing the left-truncated log-logistic (LTLL) distribution is particularly well-suited for modeling time-to-event data where observations are subject to a known lower bound such as precipitation data and cancer survival times. In this paper, we propose a Bayesian approach for estimating the parameters of the LTLL distribution with a fixed truncation point \( x_L > 0 \). Given a random variable \( X \sim LL(\alpha, \beta; x_L) \), where \( \alpha > 0 \) is the scale parameter and \( \beta > 0 \) is the shape parameter, the likelihood function is derived based on a truncated sample \( X_1, X_2, \dots, X_N \) with \( X_i > x_L \). We assume independent prior distributions for the parameters, and the posterior inference is conducted via Markov Chain Monte Carlo sampling, specifically using the Metropolis-Hastings algorithm to obtain posterior estimates \( \hat{\alpha} \) and \( \hat{\beta} \). Through simulation studies and real-world applications, we demonstrate that Bayesian estimation provides more stable and reliable parameter estimates, particularly when the likelihood surface is irregular due to left truncation. The results highlight the advantages of Bayesian inference outperform the estimation of parameter uncertainty in truncated distributions for time to event data analysis.
The introduction of the rotavirus vaccine in the United Kingdom (UK) in 2013 led to a noticeable decline in laboratory reports in subsequent years. To assess the impact of vaccination on rotavirus transmissibility we calibrated a stochastic compartmental epidemiological model using Sequential Monte Carlo (SMC) methods. Our analysis focuses on estimating the time-varying transmissibility parameter and documenting its evolution before and after vaccine rollout. We observe distinct periods of increasing and decreasing transmissibility, reflecting the dynamic response of rotavirus spread to immunization efforts. These findings improve our understanding of vaccination-driven shifts in disease transmission and provide a quantitative framework for evaluating long-term epidemiological trends.
State estimation or filtering serves as a fundamental task to enable intelligent decision-making in applications such as autonomous vehicles, robotics, healthcare monitoring, smart grids, intelligent transportation, and predictive maintenance. Standard filtering assumes prior knowledge of noise statistics to extract latent system states from noisy sensor data. However, real-world scenarios involve abnormalities like outliers, biases, drifts, and missing observations with unknown or partially known statistics, limiting conventional approaches. This thesis presents novel robust nonlinear filtering methods to mitigate these challenges. Based on insights from our filtering proposals, we extend the formulations to offline estimation/learning setups and propose smoothing extensions. Our methods leverage Bayesian inference frameworks, employing both deterministic and stochastic approximation techniques including Variational Inference (VI) and Particle Filters/Sequential Monte Carlo (SMC). We also study theoretical estimation limits using Bayesian Cram\'er-Rao bounds (BCRBs) in the context of measurement abnormalities. To validate the performance gains of the proposed methods, we perform simulations and experiments in scenarios including target tracking, indoor localization, 3D point cloud registration, mesh registration, and pose graph optimization. The fundamental nature of the work makes it useful in diverse applications, with possible future extensions toward developing outlier-robust machine learning pipelines, learning system dynamics from anomalous data, and addressing challenges in generative AI where standard diffusion models struggle with outliers, imbalanced datasets, and mode collapse.
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.
Accurately tracking an unknown and time-varying number of objects in complex environments is a significant challenge but a fundamental capability in a variety of applications, including applied ocean sciences, surveillance, autonomous driving, and wireless communications. Conventional Bayesian multiobject tracking (MOT) methods typically employ a detect-then-track (DTT) approach, where a frontend detector preprocesses raw sensor data to extract measurements for MOT. The irreversible nature of this preprocessing step can discard valuable object-related information, particularly impairing the ability to resolve weak or closely spaced objects. The track-before-detect (TBD) paradigm offers an alternative by operating directly on sensor data. However, existing TBD approaches introduce simplifications to facilitate the development of inference methods, such as assuming known signal amplitudes or conditional independence between sensor measurements given object states. These assumptions can lead to suboptimal performance and limit the applicability of the resulting TBD methods in realistic scenarios. This paper introduces coherent TBD based on a comprehensive signal model for sensor data. The new model accounts for sensor data correlations and amplitude fluctuations, enabling the accurate representation of the physics of the data-generating process in TBD. Coherent TBD is suitable for a wide range of problems in active and passive radar, active and passive sonar, as well as integrated sensing and communication systems. Based on a factor graph representation of the new measurement model, a scalable belief propagation (BP) method is developed to perform efficient Bayesian inference. Experimental results, performed with both synthetic and real data, demonstrate that the proposed method outperforms state-of-the-art conventional MOT methods.
We propose co-creative learning as a novel paradigm where humans and AI, i.e., biological and artificial agents, mutually integrate their partial perceptual information and knowledge to construct shared external representations, a process we interpret as symbol emergence. Unlike traditional AI teaching based on unilateral knowledge transfer, this addresses the challenge of integrating information from inherently different modalities. We empirically test this framework using a human-AI interaction model based on the Metropolis-Hastings naming game (MHNG), a decentralized Bayesian inference mechanism. In an online experiment, 69 participants played a joint attention naming game (JA-NG) with one of three computer agent types (MH-based, always-accept, or always-reject) under partial observability. Results show that human-AI pairs with an MH-based agent significantly improved categorization accuracy through interaction and achieved stronger convergence toward a shared sign system. Furthermore, human acceptance behavior aligned closely with the MH-derived acceptance probability. These findings provide the first empirical evidence for co-creative learning emerging in human-AI dyads via MHNG-based interaction. This suggests a promising path toward symbiotic AI systems that learn with humans, rather than from them, by dynamically aligning perceptual experiences, opening a new venue for symbiotic AI alignment.
We introduce a mathematically rigorous framework for an artificial intelligence system composed of probabilistic agents evolving through structured competition and belief revision. The architecture, grounded in Bayesian inference, measure theory, and population dynamics, defines agent fitness as a function of alignment with a fixed external oracle representing ground truth. Agents compete in a discrete-time environment, adjusting posterior beliefs through observed outcomes, with higher-rated agents reproducing and lower-rated agents undergoing extinction. Ratings are updated via pairwise truth-aligned utility comparisons, and belief updates preserve measurable consistency and stochastic convergence. We introduce hash-based cryptographic identity commitments to ensure traceability, alongside causal inference operators using do-calculus. Formal theorems on convergence, robustness, and evolutionary stability are provided. The system establishes truth as an evolutionary attractor, demonstrating that verifiable knowledge arises from adversarial epistemic pressure within a computable, self-regulating swarm.
The exponential expansion of scientific literature has surpassed the epistemic processing capabilities of both human experts and current artificial intelligence systems. This paper introduces Bayesian Epistemology with Weighted Authority (BEWA), a formally structured architecture that operationalises belief as a dynamic, probabilistically coherent function over structured scientific claims. Each claim is contextualised, author-attributed, and evaluated through a system of replication scores, citation weighting, and temporal decay. Belief updates are performed via evidence-conditioned Bayesian inference, contradiction processing, and epistemic decay mechanisms. The architecture supports graph-based claim propagation, authorial credibility modelling, cryptographic anchoring, and zero-knowledge audit verification. By formalising scientific reasoning into a computationally verifiable epistemic network, BEWA advances the foundation for machine reasoning systems that promote truth utility, rational belief convergence, and audit-resilient integrity across dynamic scientific domains.
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.
Real-time identification and quantification of greenhouse-gas emissions under transient atmospheric conditions is a critical challenge in environmental monitoring. We introduce a spatio-temporal inversion framework that embeds a deep-learning surrogate of computational fluid dynamics (CFD) within a sequential Monte Carlo algorithm to perform Bayesian inference of both emission rate and source location in dynamic flow fields. By substituting costly numerical solvers with a multilayer perceptron trained on high-fidelity CFD outputs, our surrogate captures spatial heterogeneity and temporal evolution of gas dispersion, while delivering near-real-time predictions. Validation on the Chilbolton methane release dataset demonstrates comparable accuracy to full CFD solvers and Gaussian plume models, yet achieves orders-of-magnitude faster runtimes. Further experiments under simulated obstructed-flow scenarios confirm robustness in complex environments. This work reconciles physical fidelity with computational feasibility, offering a scalable solution for industrial emissions monitoring and other time-sensitive spatio-temporal inversion tasks in environmental and scientific modeling.
In-context learning (ICL) has emerged as a powerful paradigm for task adaptation in large language models (LLMs), where models infer underlying task structures from a few demonstrations. However, ICL remains susceptible to biases that arise from prior knowledge and contextual demonstrations, which can degrade the performance of LLMs. Existing bias calibration methods typically apply fixed class priors across all inputs, limiting their efficacy in dynamic ICL settings where the context for each query differs. To address these limitations, we adopt implicit sequential Bayesian inference as a framework for interpreting ICL, identify "surprise" as an informative signal for class prior shift, and introduce a novel method--Surprise Calibration (SC). SC leverages the notion of surprise to capture the temporal dynamics of class priors, providing a more adaptive and computationally efficient solution for in-context learning. We empirically demonstrate the superiority of SC over existing bias calibration techniques across a range of benchmark natural language processing tasks.
Accurate and interpretable car-following models are essential for traffic simulation and autonomous vehicle development. However, classical models like the Intelligent Driver Model (IDM) are fundamentally limited by their parsimonious and single-regime structure. They fail to capture the multi-modal nature of human driving, where a single driving state (e.g., speed, relative speed, and gap) can elicit many different driver actions. This forces the model to average across distinct behaviors, reducing its fidelity and making its parameters difficult to interpret. To overcome this, we introduce a regime-switching framework that allows driving behavior to be governed by different IDM parameter sets, each corresponding to an interpretable behavioral mode. This design enables the model to dynamically switch between interpretable behavioral modes, rather than averaging across diverse driving contexts. We instantiate the framework using a Factorial Hidden Markov Model with IDM dynamics (FHMM-IDM), which explicitly separates intrinsic driving regimes (e.g., aggressive acceleration, steady-state following) from external traffic scenarios (e.g., free-flow, congestion, stop-and-go) through two independent latent Markov processes. Bayesian inference via Markov chain Monte Carlo (MCMC) is used to jointly estimate the regime-specific parameters, transition dynamics, and latent state trajectories. Experiments on the HighD dataset demonstrate that FHMM-IDM uncovers interpretable structure in human driving, effectively disentangling internal driver actions from contextual traffic conditions and revealing dynamic regime-switching patterns. This framework provides a tractable and principled solution to modeling context-dependent driving behavior under uncertainty, offering improvements in the fidelity of traffic simulations, the efficacy of safety analyses, and the development of more human-centric ADAS.
Research on lane change prediction has gained a lot of momentum in the last couple of years. However, most research is confined to simulation or results obtained from datasets, leaving a gap between algorithmic advances and on-road deployment. This work closes that gap by demonstrating, on real hardware, a lane-change prediction system based on Knowledge Graph Embeddings (KGEs) and Bayesian inference. Moreover, the ego-vehicle employs a longitudinal braking action to ensure the safety of both itself and the surrounding vehicles. Our architecture consists of two modules: (i) a perception module that senses the environment, derives input numerical features, and converts them into linguistic categories; and communicates them to the prediction module; (ii) a pretrained prediction module that executes a KGE and Bayesian inference model to anticipate the target vehicle's maneuver and transforms the prediction into longitudinal braking action. Real-world hardware experimental validation demonstrates that our prediction system anticipates the target vehicle's lane change three to four seconds in advance, providing the ego vehicle sufficient time to react and allowing the target vehicle to make the lane change safely.
Varying coefficient models are popular for estimating nonlinear regression functions in functional data models. Their Bayesian variants have received limited attention in large data applications, primarily due to prohibitively slow posterior computations using Markov chain Monte Carlo (MCMC) algorithms. We introduce Bayesian data sketching for varying coefficient models to obviate computational challenges presented by large sample sizes. To address the challenges of analyzing large data, we compress the functional response vector and predictor matrix by a random linear transformation to achieve dimension reduction and conduct inference on the compressed data. Our approach distinguishes itself from several existing methods for analyzing large functional data in that it requires neither the development of new models or algorithms, nor any specialized computational hardware while delivering fully model-based Bayesian inference. Well-established methods and algorithms for varying coefficient regression models can be applied to the compressed data.