Bayesian Deep Learning

With the development of biotechnology, RNA therapies have shown great potential. However, different from proteins, the sequences corresponding to a single RNA three-dimensional structure are more abundant. Most of the existing RNA design methods merely take into account the secondary structure of RNA, or are only capable of generating a limited number of candidate sequences. To address these limitations, we propose a geometric-algebra-enhanced $\textbf{B}$ayesian $\textbf{F}$low $\textbf{N}$etwork for the inverse design of $\textbf{R}$NA, called $\textbf{RBFN}$. RBFN uses a Bayesian Flow Network to model the distribution of nucleotide sequences in RNA, enabling the generation of more reasonable RNA sequences. Meanwhile, considering the more flexible characteristics of RNA conformations, we utilize geometric algebra to enhance the modeling ability of the RNA three-dimensional structure, facilitating a better understanding of RNA structural properties. In addition, due to the scarcity of RNA structures and the limitation that there are only four types of nucleic acids, we propose a new time-step distribution sampling to address the scarcity of RNA structure data and the relatively small number of nucleic acid types. Evaluation on the single-state fixed-backbone re-design benchmark and multi-state fixed-backbone benchmark indicates that RBFN can outperform existing RNA design methods in various RNA design tasks, enabling effective RNA sequence design.

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.

This work establishes that sparse Bayesian neural networks achieve optimal posterior contraction rates over anisotropic Besov spaces and their hierarchical compositions. These structures reflect the intrinsic dimensionality of the underlying function, thereby mitigating the curse of dimensionality. Our analysis shows that Bayesian neural networks equipped with either sparse or continuous shrinkage priors attain the optimal rates which are dependent on the intrinsic dimension of the true structures. Moreover, we show that these priors enable rate adaptation, allowing the posterior to contract at the optimal rate even when the smoothness level of the true function is unknown. The proposed framework accommodates a broad class of functions, including additive and multiplicative Besov functions as special cases. These results advance the theoretical foundations of Bayesian neural networks and provide rigorous justification for their practical effectiveness in high-dimensional, structured estimation problems.

Squared families of probability distributions have been studied and applied in numerous machine learning contexts. Typically, they appear as likelihoods, where their advantageous computational, geometric and statistical properties are exploited for fast estimation algorithms, representational properties and statistical guarantees. Here, we investigate the use of squared families as prior beliefs in Bayesian inference. We find that they can form helpful conjugate families, often allowing for closed-form and tractable Bayesian inference and marginal likelihoods. We apply such conjugate families to Bayesian regression in feature space using end-to-end learnable neural network features. Such a setting allows for a rich multi-modal alternative to Gaussian processes with neural network features, often called deep kernel learning. We demonstrate our method on few shot learning, outperforming existing neural methods based on Gaussian processes and normalising flows.

Variational mean field approximations tend to struggle with contemporary overparametrized deep neural networks. Where a Bayesian treatment is usually associated with high-quality predictions and uncertainties, the practical reality has been the opposite, with unstable training, poor predictive power, and subpar calibration. Building upon recent work on reparametrizations of neural networks, we propose a simple variational family that considers two independent linear subspaces of the parameter space. These represent functional changes inside and outside the support of training data. This allows us to build a fully-correlated approximate posterior reflecting the overparametrization that tunes easy-to-interpret hyperparameters. We develop scalable numerical routines that maximize the associated evidence lower bound (ELBO) and sample from the approximate posterior. Empirically, we observe state-of-the-art performance across tasks, models, and datasets compared to a wide array of baseline methods. Our results show that approximate Bayesian inference applied to deep neural networks is far from a lost cause when constructing inference mechanisms that reflect the geometry of reparametrizations.

Dynamic pricing algorithms typically assume continuous price variables, which may not reflect real-world scenarios where prices are often discrete. This paper demonstrates that leveraging discrete price information within a semi-parametric model can substantially improve performance, depending on the size of the support set of the price variable relative to the time horizon. Specifically, we propose a novel semi-parametric contextual dynamic pricing algorithm, namely BayesCoxCP, based on a Bayesian approach to the Cox proportional hazards model. Our theoretical analysis establishes high-probability regret bounds that adapt to the sparsity level $\gamma$, proving that our algorithm achieves a regret upper bound of $\widetilde{O}(T^{(1+\gamma)/2}+\sqrt{dT})$ for $\gamma < 1/3$ and $\widetilde{O}(T^{2/3}+\sqrt{dT})$ for $\gamma \geq 1/3$, where $\gamma$ represents the sparsity of the price grid relative to the time horizon $T$. Through numerical experiments, we demonstrate that our proposed algorithm significantly outperforms an existing method, particularly in scenarios with sparse discrete price points.

Metal-Organic Frameworks (MOFs) have attracted considerable attention due to their unique properties including high surface area and tunable porosity, and promising applications in catalysis, gas storage, and drug delivery. Structure prediction for MOFs is a challenging task, as these frameworks are intrinsically periodic and hierarchically organized, where the entire structure is assembled from building blocks like metal nodes and organic linkers. To address this, we introduce MOF-BFN, a novel generative model for MOF structure prediction based on Bayesian Flow Networks (BFNs). Given the local geometry of building blocks, MOF-BFN jointly predicts the lattice parameters, as well as the positions and orientations of all building blocks within the unit cell. In particular, the positions are modelled in the fractional coordinate system to naturally incorporate the periodicity. Meanwhile, the orientations are modeled as unit quaternions sampled from learned Bingham distributions via the proposed Bingham BFN, enabling effective orientation generation on the 4D unit hypersphere. Experimental results demonstrate that MOF-BFN achieves state-of-the-art performance across multiple tasks, including structure prediction, geometric property evaluation, and de novo generation, offering a promising tool for designing complex MOF materials.

Deep Neural Networks (DNNs) have demonstrated remarkable performance across various domains, including computer vision and natural language processing. However, they often struggle to accurately quantify their predictions' uncertainty, limiting their broader adoption in critical industrial applications. Uncertainty Quantification (UQ) for Deep Learning seeks to address this challenge by providing methodologies to improve the reliability of uncertainty estimates. While numerous techniques have been proposed, a unified tool remains lacking that offers a seamless workflow for evaluating and integrating these methods. To bridge this gap, we introduce **Torch-Uncertainty**, a *PyTorch* and *Lightning* framework designed to streamline the training and evaluation of DNNs with UQ techniques. In this paper, we outline the foundational principles of our library and present comprehensive experimental results that benchmark a diverse set of UQ methods across classification, segmentation, and regression tasks. Our library is available at: https://github.com/ENSTA-U2IS-AI/torch-uncertainty.

Concept Bottleneck Models (CBMs) have been proposed as a compromise between white-box and black-box models, aiming to achieve interpretability without sacrificing accuracy. The standard training procedure for CBMs is to predefine a candidate set of human-interpretable concepts, extract their values from the training data, and identify a sparse subset as inputs to a transparent prediction model. However, such approaches are often hampered by the tradeoff between exploring a sufficiently large set of concepts versus controlling the cost of obtaining concept extractions, resulting in a large interpretability-accuracy tradeoff. This work investigates a novel approach that sidesteps these challenges: BC-LLM iteratively searches over a potentially infinite set of concepts within a Bayesian framework, in which Large Language Models (LLMs) serve as both a concept extraction mechanism and prior. Even though LLMs can be miscalibrated and hallucinate, we prove that BC-LLM can provide rigorous statistical inference and uncertainty quantification. Across image, text, and tabular datasets, BC-LLM outperforms interpretable baselines and even black-box models in certain settings, converges more rapidly towards relevant concepts, and is more robust to out-of-distribution samples.

Estimating the uncertainty of responses from Large Language Models (LLMs) remains a critical challenge. While recent Bayesian methods have demonstrated effectiveness in quantifying uncertainty through low-rank weight updates, they typically require complex fine-tuning or post-training procedures. In this paper, we propose **T**raining-**F**ree **B**ayesianization (**TFB**), a simple yet theoretically grounded framework that efficiently transforms trained low-rank adapters into Bayesian ones without additional training. TFB systematically searches for the maximally acceptable level of variance in the weight posterior, constrained within a family of low-rank isotropic Gaussian distributions. Our theoretical analysis shows that under mild conditions, this search process is equivalent to KL-regularized variational optimization, a generalized form of variational inference. Through comprehensive experiments, we show that TFB achieves superior uncertainty estimation and generalization compared to existing methods while eliminating the need for complex Bayesianization training procedures.

Efficient online decision-making in contextual bandits is challenging, as methods without informative priors often suffer from computational or statistical inefficiencies. In this work, we leverage pre-trained diffusion models as expressive priors to capture complex action dependencies and develop a practical algorithm that efficiently approximates posteriors under such priors, enabling both fast updates and sampling. Empirical results demonstrate the effectiveness and versatility of our approach across diverse contextual bandit settings.

Programmatic Weak Supervision (PWS) aims to reduce the cost of constructing large high quality labeled datasets often used in training modern machine learning models. A major component of the PWS pipeline is the label model, which amalgamates predictions from multiple noisy weak supervision sources, i.e. labeling functions (LFs), to label datapoints. While most label models are either probabilistic or adversarial, a recently proposed label model achieves strong empirical performance without falling into either camp. That label model constructs a polytope of plausible labelings using the LF predictions and outputs the "center" of that polytope as its proposed labeling. In this paper, we attempt to theoretically study that strategy by proposing Bayesian Balsubramani-Freund (BBF), a label model that implicitly constructs a polytope of plausible labelings and selects a labeling in its interior. We show an assortment of statistical results for BBF: log-concavity of its posterior, its form of solution, consistency, and rates of convergence. Extensive experiments compare our proposed method against twelve baseline label models over eleven datasets. BBF compares favorably to other Bayesian label models and label models that don't use datapoint features -- matching or exceeding their performance on eight out of eleven datasets.

In this work, we introduce TreeGen, a novel generative framework modeling distributions over hierarchies. We extend Bayesian Flow Networks (BFNs) to enable transitions between probabilistic and discrete hierarchies parametrized via categorical distributions. Our proposed scheduler provides smooth and consistent entropy decay across varying numbers of categories. We empirically evaluate TreeGen on the jet-clustering task in high-energy physics, demonstrating that it consistently generates valid trees that adhere to physical constraints and closely align with ground-truth log-likelihoods. Finally, by comparing TreeGen’s samples to the exact posterior distribution and performing likelihood maximization via rejection sampling, we demonstrate that TreeGen outperforms various baselines.

We introduce *NeuralSurv*, the first deep survival model to incorporate Bayesian uncertainty quantification. Our non‑parametric, architecture‑agnostic framework flexibly captures time‑varying covariate–risk relationships in continuous time via a novel two‑stage data‑augmentation scheme, for which we establish theoretical guarantees. For efficient posterior inference, we introduce a mean‑field variational algorithm with coordinate‑ascent updates that scale linearly in model size. By locally linearizing the Bayesian neural network, we obtain full conjugacy and derive all coordinate updates in closed form. In experiments, *NeuralSurv* delivers superior calibration compared to state-of-the-art deep survival models, while matching or exceeding their discriminative performance across both synthetic benchmarks and real-world datasets. Our results demonstrate the value of Bayesian principles in data‑scarce regimes by enhancing model calibration and providing robust, well‑calibrated uncertainty estimates for the survival function.

Many domain experts do not have the time or expertise to write formal Bayesian models. This paper takes an informal problem description as input, and combines a large language model and a probabilistic programming language to define a joint distribution over formal models, latent variables, and data. A posterior over latent variables follows by conditioning on observed data and integrating over formal models. This presents a challenging inference problem. We suggest an inference recipe that amounts to generating many formal models from the large language model, performing approximate inference on each, and then doing a weighted aver- age. This is justified and analyzed as a combination of self-normalized importance sampling, MCMC, and importance-weighted variational inference. Experimentally, this produces sensible predictions from only data and an informal problem description, without the need to specify a formal model.

This paper introduces the Quotient Bayesian Learning Rule, an extension of natural-gradient Bayesian updates to probability models that fall outside the exponential family. Building on the observation that many heavy-tailed and otherwise non-exponential distributions arise as marginals of minimal exponential families, we prove that such marginals inherit a unique Fisher–Rao information geometry via the quotient-manifold construction. Exploiting this geometry, we derive the Quotient Natural Gradient algorithm, which takes steepest-descent steps in the well-structured covering space, thereby guaranteeing parameterization-invariant optimization in the target space. Empirical results on the Student-$t$ distribution confirm that our method converges more rapidly and attains higher-quality solutions than previous variants of the Bayesian Learning Rule. These findings position quotient geometry as a unifying tool for efficient and principled inference across a broad class of latent-variable models.

Personalized Bayesian federated learning (PBFL) handles non-i.i.d. client data and quantifies uncertainty by combining personalization with Bayesian inference. However, current PBFL methods face two main limitations: posterior inference on clients often assumes restrictive parametric forms, and server-side posterior aggregation typically relies on naive parameter averaging. To overcome these issues, we propose FedWBA, a novel PBFL method that enhances both local inference and global aggregation. At the client level, we use particle-based variational inference for nonparametric posterior representation. At the server level, we introduce particle-based Wasserstein barycenter aggregation, offering a more geometrically meaningful approach. Theoretically, we provide local and global convergence guarantees for FedWBA. Locally, we prove a KL divergence decrease lower bound per iteration for variational inference convergence. Globally, we show that the Wasserstein barycenter converges to the true parameter as the client data size increases. Empirically, experiments show that FedWBA outperforms baselines in prediction accuracy, uncertainty calibration, and convergence rate, with ablation studies confirming its robustness.

We initiate the study of multidimensional Bayesian utility maximization, focusing on the unit-demand setting where values are i.i.d. across both items and buyers. The seminal result of Hartline and Roughgarden '08 studies simple, information-robust mechanisms that maximize utility for $n$ i.i.d. agents and $m$ identical items via an approximation to social welfare as an upper bound, and they prove this gap between optimal utility and social welfare is $\Theta(1+\log{n/m})$ in this setting. We extend these results to the multidimensional setting. To do so, we develop simple, prior-independent, approximately-optimal mechanisms, targeting the simplest benchmark of optimal welfare. We give a $(1-1/e)$-approximation when there are more items than buyers, and a $\Theta(\log{n/m})$-approximation when there are more buyers than items, and we prove that this bound is tight in both $n$ and $m$ by reducing the i.i.d. unit-demand setting to the identical items setting. Finally, we include an extensive discussion section on why Bayesian utility maximization is a promising research direction. In particular, we characterize complexities in this setting that defy our intuition from the welfare and revenue literature, and motivate why coming up with a better benchmark than welfare is a hard problem itself.

While neural networks have demonstrated impressive performance across various tasks, accurately quantifying uncertainty in their predictions is essential to ensure their trustworthiness and enable widespread adoption in critical systems. Several Bayesian uncertainty quantification (UQ) methods exist that are either cheap or reliable, but not both. We propose a post-hoc, sampling-based UQ method for overparameterized networks at the end of training. Our approach constructs efficient and meaningful deep ensembles by employing a (stochastic) gradient-descent sampling process on appropriately linearized networks. We demonstrate that our method effectively approximates the posterior of a Gaussian Process using the empirical Neural Tangent Kernel. Through a series of numerical experiments, we show that our method not only outperforms competing approaches in computational efficiency--often reducing costs by multiple factors--but also maintains state-of-the-art performance across a variety of UQ metrics for both regression and classification tasks.

In Bayesian persuasion, an informed sender strategically discloses information to a receiver so as to persuade them to undertake desirable actions. Recently, Markov persuasion processes (MPPs) have been introduced to capture sequential scenarios where a sender faces a stream of myopic receivers in a Markovian environment. The MPPs studied so far in the literature suffer from issues that prevent them from being fully operational in practice, e.g., they assume that the sender knows receivers' rewards. We fix such issues by addressing MPPs where the sender has no knowledge about the environment. We design a learning algorithm for the sender, working with partial feedback. We prove that its regret with respect to an optimal information-disclosure policy grows sublinearly in the number of episodes, as it is the case for the loss in persuasiveness cumulated while learning. Moreover, we provide lower bounds for our setting matching the guarantees of our algorithm.