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.
Variational inference (VI) is a popular method for approximating intractable posterior distributions in Bayesian inference and probabilistic machine learning. In this paper, we introduce a general framework for quantifying the statistical accuracy of mean-field variational inference (MFVI) for posterior approximation in Bayesian latent variable models with categorical local latent variables. Utilizing our general framework, we capture the exact asymptotic regime where MFVI `works' for the celebrated latent Dirichlet allocation (LDA) model. Focusing on the mixed membership stochastic blockmodel (MMSB), we show that the vanilla fully factorized MFVI, often used in the literature, is suboptimal. We propose a partially grouped VI algorithm for this model and show that it works, and derive its exact asymptotic performance. We further illustrate that our bounds are tight for both the above models.
Mean-field variational inference (MFVI) is a widely used method for approximating high-dimensional probability distributions by product measures. This paper studies the stability properties of the mean-field approximation when the target distribution varies within the class of strongly log-concave measures. We establish dimension-free Lipschitz continuity of the MFVI optimizer with respect to the target distribution, measured in the 2-Wasserstein distance, with Lipschitz constant inversely proportional to the log-concavity parameter. Under additional regularity conditions, we further show that the MFVI optimizer depends differentiably on the target potential and characterize the derivative by a partial differential equation. Methodologically, we follow a novel approach to MFVI via linearized optimal transport: the non-convex MFVI problem is lifted to a convex optimization over transport maps with a fixed base measure, enabling the use of calculus of variations and functional analysis. We discuss several applications of our results to robust Bayesian inference and empirical Bayes, including a quantitative Bernstein--von Mises theorem for MFVI, as well as to distributed stochastic control.
In this paper, we consider uplink channel estimation for massive multi-input multi-output (MIMO) systems with partially connected hybrid beamforming (PC-HBF) structures. Existing beam design and channel estimation schemes are usually based on ideal assumptions and require transmitting pilots across multiple timeslots, making them unsuitable for practical PC-HBF systems. To overcome these drawbacks, we propose a novel beam design and a corresponding channel estimation algorithm to achieve accurate and real-time uplink channel estimation. Firstly, we introduce a group-wise narrow beam design in the vertical dimension to suppress interference from undesired angular components and improve vertical angle estimation accuracy,which divides the columns of the uniform planar array (UPA)into groups and the vertical angle interval into sub-intervals.In this way, each group is assigned with a narrow beam to cover one vertical angle sub-interval, and the set of narrow beams is designed based on the filter design theory. Secondly, we optimize the antenna grouping pattern using the Estimation of Distribution Algorithm (EDA), balancing interference suppression and resolution capability in the horizontal dimension, leading to a better horizontal angle estimation performance. Finally, we design a low-complexity group-wise subspace constrained variational Bayesian inference (GW-SC-VBI) algorithm to fully take advantage of the proposed beam design to achieve both low-complexity and high-accurate channel estimation. Simulation results demonstrate that the proposed scheme achieves notable performance gains over baseline methods.
Massive connectivity supports the sporadic access of a vast number of devices without requiring prior permission from the base station (BS). In such scenarios, the BS must perform joint activity detection and channel estimation (JADCE) prior to data reception. Message passing algorithms have emerged as a prominent solution for JADCE under a Bayesian inference framework. The existing message passing algorithms, however, typically rely on some hand-crafted and overly simplistic priors of the wireless channel, leading to significant channel estimation errors and reduced activity detection accuracy. In this paper, we focus on the problem of JADCE in a multiple-input multiple-output orthogonal frequency division multiplexing (MIMO-OFDM) grant-free random access network. We propose to incorporate a more accurate channel prior learned by score-based generative models into message passing, so as to push towards the performance limit of JADCE. Specifically, we develop a novel turbo message passing (TMP) framework that models the entire channel matrix as a super node, rather than factorizing it element-wise. This design enables the seamless integration of score-based generative models as a minimum mean-squared error (MMSE) denoiser. The variance of the denoiser, which is essential in message passing, can also be learned through score-based generative models. Our approach, termed score-based TMP for JADCE (STMP-JADCE), takes full advantages of the powerful generative prior and, meanwhile, benefits from the fast convergence speed of message passing. Numerical simulations show that STMP-JADCE drastically enhances the activity detection and channel estimation performance compared to the state-of-the-art baseline algorithms.
High-resolution soil moisture (SM) observations are critical for agricultural monitoring, forestry management, and hazard prediction, yet current satellite passive microwave missions cannot directly provide retrievals at tens-of-meter spatial scales. Unmanned aerial vehicle (UAV) mounted microwave radiometry presents a promising alternative, but most evaluations to date have focused on agricultural settings, with limited exploration across other land covers and few efforts to quantify retrieval uncertainty. This study addresses both gaps by evaluating SM retrievals from a drone-based Portable L-band Radiometer (PoLRa) across shrubland, bare soil, and forest strips in Central Illinois, U.S., using a 10-day field campaign in 2024. Controlled UAV flights at altitudes of 10 m, 20 m, and 30 m were performed to generate brightness temperatures (TB) at spatial resolutions of 7 m, 14 m, and 21 m. SM retrievals were carried out using multiple tau-omega-based algorithms, including the single channel algorithm (SCA), dual channel algorithm (DCA), and multi-temporal dual channel algorithm (MTDCA). A Bayesian inference framework was then applied to provide probabilistic uncertainty characterization for both SM and vegetation optical depth (VOD). Results show that the gridded TB distributions consistently capture dry-wet gradients associated with vegetation density variations, and spatial correlations between polarized observations are largely maintained across scales. Validation against in situ measurements indicates that PoLRa derived SM retrievals from the SCAV and MTDCA algorithms achieve unbiased root-mean-square errors (ubRMSE) generally below 0.04 m3/m3 across different land covers. Bayesian posterior analyses confirm that reference SM values largely fall within the derived uncertainty intervals, with mean uncertainty ranges around 0.02 m3/m3 and 0.11 m3/m3 for SCA and DCA related retrievals.
Binary Neural Networks (BiNNs), which employ single-bit precision weights, have emerged as a promising solution to reduce memory usage and power consumption while maintaining competitive performance in large-scale systems. However, training BiNNs remains a significant challenge due to the limitations of conventional training algorithms. Quantum HyperNetworks offer a novel paradigm for enhancing the optimization of BiNN by leveraging quantum computing. Specifically, a Variational Quantum Algorithm is employed to generate binary weights through quantum circuit measurements, while key quantum phenomena such as superposition and entanglement facilitate the exploration of a broader solution space. In this work, we establish a connection between this approach and Bayesian inference by deriving the Evidence Lower Bound (ELBO), when direct access to the output distribution is available (i.e., in simulations), and introducing a surrogate ELBO based on the Maximum Mean Discrepancy (MMD) metric for scenarios involving implicit distributions, as commonly encountered in practice. Our experimental results demonstrate that the proposed methods outperform standard Maximum Likelihood Estimation (MLE), improving trainability and generalization.
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.
Matrix factorization is a widely used technique in recommendation systems. Probabilistic Matrix Factorization (PMF) [1] extends traditional matrix factorization by incorporating probability distributions over latent factors, allowing for uncertainty quantification. However, computing the posterior distribution is intractable due to the high-dimensional integral. To address this, we employ two Bayesian inference methods: Markov Chain Monte Carlo (MCMC) [2] and Variational Inference (VI) [3] to approximate the posterior. We evaluate their performance on MovieLens dataset and compare their convergence speed, predictive accuracy, and computational efficiency. Experimental results demonstrate that VI offers faster convergence, while MCMC provides more accurate posterior estimates.
Prior-data fitted networks (PFNs) have recently been proposed as a promising way to train tabular foundation models. PFNs are transformers that are pre-trained on synthetic data generated from a prespecified prior distribution and that enable Bayesian inference through in-context learning. In this paper, we introduce CausalFM, a comprehensive framework for training PFN-based foundation models in various causal inference settings. First, we formalize the construction of Bayesian priors for causal inference based on structural causal models (SCMs) in a principled way and derive necessary criteria for the validity of such priors. Building on this, we propose a novel family of prior distributions using causality-inspired Bayesian neural networks that enable CausalFM to perform Bayesian causal inference in various settings, including back-door, front-door, and instrumental variable adjustment. Finally, we instantiate CausalFM and explicitly train a foundation model for estimating conditional average treatment effects (CATEs) using back-door adjustment. We show that CausalFM performs competitively for CATE estimation using various synthetic and semi-synthetic benchmarks. In sum, our framework can be used as a general recipe to train foundation models for various causal inference settings. In contrast to the current state-of-the-art in causal inference, CausalFM offers a novel paradigm with the potential to fundamentally change how practitioners perform causal inference in medicine, economics, and other disciplines.
Although large language models (LLMs) are becoming increasingly capable of solving challenging real-world tasks, accurately quantifying their uncertainty remains a critical open problem, which limits their applicability in high-stakes domains. This challenge is further compounded by the closed-source, black-box nature of many state-of-the-art LLMs. Moreover, LLM-based systems can be highly sensitive to the prompts that bind them together, which often require significant manual tuning (i.e., prompt engineering). In this work, we address these challenges by viewing LLM-based systems through a Bayesian lens. We interpret prompts as textual parameters in a statistical model, allowing us to use a small training dataset to perform Bayesian inference over these prompts. This novel perspective enables principled uncertainty quantification over both the model's textual parameters and its downstream predictions, while also incorporating prior beliefs about these parameters expressed in free-form text. To perform Bayesian inference, a difficult problem even for well-studied data modalities, we introduce Metropolis-Hastings through LLM Proposals (MHLP), a novel Markov chain Monte Carlo (MCMC) algorithm that combines prompt optimization techniques with standard MCMC methods. MHLP is a turnkey modification to existing LLM pipelines, including those that rely exclusively on closed-source models. Empirically, we demonstrate that our method yields improvements in both predictive accuracy and uncertainty quantification (UQ) on a range of LLM benchmarks and UQ tasks. More broadly, our work demonstrates a viable path for incorporating methods from the rich Bayesian literature into the era of LLMs, paving the way for more reliable and calibrated LLM-based systems.
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 conventional FL methods face critical challenges in realistic wireless edge networks, where training data is both limited and heterogeneous, often leading to unstable training and poor generalization. To address these challenges in a principled manner, we propose a novel wireless FL framework grounded in Bayesian inference. By virtue of the Bayesian approach, our framework captures model uncertainty by maintaining distributions over local weights and performs distribution-level aggregation of local distributions into a global distribution. This mitigates local overfitting and client drift, thereby enabling more reliable inference. Nevertheless, adopting Bayesian FL increases communication overhead due to the need to transmit richer model information and fundamentally alters the aggregation process beyond simple averaging. As a result, conventional Over-the-Air Computation (AirComp), widely used to improve communication efficiency in standard FL, is no longer directly applicable. To overcome this limitation, we design a dedicated AirComp scheme tailored to Bayesian FL, which efficiently aggregates local posterior distributions at the distribution level by exploiting the superposition property of wireless channels. In addition, we derive an optimal transmit power control strategy, grounded in rigorous convergence analysis, to accelerate training under power constraints. Our analysis explicitly accounts for practical wireless impairments such as fading and noise, and provides theoretical guarantees for convergence. Extensive simulations validate the proposed framework, demonstrating significant improvements in test accuracy and calibration performance over conventional FL methods, particularly in data-scarce and heterogeneous environments.
Real-world robotic applications, from autonomous exploration to assistive technologies, require adaptive, interpretable, and data-efficient learning paradigms. While deep learning architectures and foundation models have driven significant advances in diverse robotic applications, they remain limited in their ability to operate efficiently and reliably in unknown and dynamic environments. In this position paper, we critically assess these limitations and introduce a conceptual framework for combining data-driven learning with deliberate, structured reasoning. Specifically, we propose leveraging differentiable physics for efficient world modeling, Bayesian inference for uncertainty-aware decision-making, and meta-learning for rapid adaptation to new tasks. By embedding physical symbolic reasoning within neural models, robots could generalize beyond their training data, reason about novel situations, and continuously expand their knowledge. We argue that such hybrid neuro-symbolic architectures are essential for the next generation of autonomous systems, and to this end, we provide a research roadmap to guide and accelerate their development.
Automated evaluation of free-form outputs from large language models (LLMs) is challenging because many distinct answers can be equally valid. A common practice is to use LLMs themselves as judges, but the theoretical properties of this approach are not yet well understood. We show that a geometric framework that represents both judges and candidates as points on a probability simplex can provide helpful insight on what is or is not identifiable using LLM judges. Our theoretical analysis uncovers a "phase transition" in ranking identifiability: for binary scoring systems, true rankings are identifiable even with weak judges under mild assumptions, while rankings become non-identifiable for three or more scoring levels even with infinite data, absent additional prior knowledge. This non-identifiability highlights how uncertainty in rankings stems from not only aleatoric uncertainty (i.e., inherent stochasticity in the data) but also epistemic uncertainty regarding which assumptions hold, an aspect that has received limited attention until now. To integrate both types of uncertainty, we use Bayesian inference to encode assumptions as priors and conduct sensitivity analysis of ranking estimates and credible intervals. Empirical evaluations across multiple benchmarks demonstrate that Bayesian inference yields more accurate rankings and substantially improves coverage rates. These results underscore the importance of taking a more holistic approach to uncertainty quantification when using LLMs as judges.
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.
Variational inference often struggles with the posterior geometry exhibited by complex hierarchical Bayesian models. Recent advances in flow-based variational families and Variationally Inferred Parameters (VIP) each address aspects of this challenge, but their formal relationship is unexplored. Here, we prove that the combination of VIP and a full-rank Gaussian can be represented exactly as a forward autoregressive flow augmented with a translation term and input from the model's prior. Guided by this theoretical insight, we introduce the Model-Informed Flow (MIF) architecture, which adds the necessary translation mechanism, prior information, and hierarchical ordering. Empirically, MIF delivers tighter posterior approximations and matches or exceeds state-of-the-art performance across a suite of hierarchical and non-hierarchical benchmarks.
Training neural networks on randomly generated artificial datasets yields Bayesian models that capture the prior defined by the dataset-generating distribution. Prior-data Fitted Networks (PFNs) are a class of methods designed to leverage this insight. In an era of rapidly increasing computational resources for pre-training and a near stagnation in the generation of new real-world data in many applications, PFNs are poised to play a more important role across a wide range of applications. They enable the efficient allocation of pre-training compute to low-data scenarios. Originally applied to small Bayesian modeling tasks, the field of PFNs has significantly expanded to address more complex domains and larger datasets. This position paper argues that PFNs and other amortized inference approaches represent the future of Bayesian inference, leveraging amortized learning to tackle data-scarce problems. We thus believe they are a fruitful area of research. In this position paper, we explore their potential and directions to address their current limitations.
Language models exhibit complex, diverse behaviors when prompted with free-form text, making it difficult to characterize the space of possible outputs. We study the problem of behavior elicitation, where the goal is to search for prompts that induce specific target behaviors (e.g., hallucinations or harmful responses) from a target language model. To navigate the exponentially large space of possible prompts, we train investigator models to map randomly-chosen target behaviors to a diverse distribution of outputs that elicit them, similar to amortized Bayesian inference. We do this through supervised fine-tuning, reinforcement learning via DPO, and a novel Frank-Wolfe training objective to iteratively discover diverse prompting strategies. Our investigator models surface a variety of effective and human-interpretable prompts leading to jailbreaks, hallucinations, and open-ended aberrant behaviors, obtaining a 100% attack success rate on a subset of AdvBench (Harmful Behaviors) and an 85% hallucination rate.
We develop a new algorithm for inference based on structural vector autoregressions (SVARs) identified with sign restrictions. The key insight of our algorithm is to break apart from the accept-reject tradition associated with sign-identified SVARs. We show that embedding an elliptical slice sampling within a Gibbs sampler approach can deliver dramatic gains in speed and turn previously infeasible applications into feasible ones. We provide a tractable example to illustrate the power of the elliptical slice sampling applied to sign-identified SVARs. We demonstrate the usefulness of our algorithm by applying it to a well-known small-SVAR model of the oil market featuring a tight identified set, as well as to a large SVAR model with more than 100 sign restrictions.