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We present a new sampling-based approach for enabling efficient computation of low-rank Bayesian matrix completion and quantifying the associated uncertainty. Firstly, we design a new prior model based on the singular-value-decomposition (SVD) parametrization of low-rank matrices. Our prior is analogous to the seminal nuclear-norm regularization used in non-Bayesian setting and enforces orthogonality in the factor matrices by constraining them to Stiefel manifolds. Then, we design a geodesic Hamiltonian Monte Carlo (-within-Gibbs) algorithm for generating posterior samples of the SVD factor matrices. We demonstrate that our approach resolves the sampling difficulties encountered by standard Gibbs samplers for the common two-matrix factorization used in matrix completion. More importantly, the geodesic Hamiltonian sampler allows for sampling in cases with more general likelihoods than the typical Gaussian likelihood and Gaussian prior assumptions adopted in most of the existing Bayesian matrix completion literature. We demonstrate an applications of our approach to fit the categorical data of a mice protein dataset and the MovieLens recommendation problem. Numerical examples demonstrate superior sampling performance, including better mixing and faster convergence to a stationary distribution. Moreover, they demonstrate improved accuracy on the two real-world benchmark problems we considered.

Recent advancements in Large Language Models (LLMs) are transforming biology, computer science, and many other research fields, as well as impacting everyday life. While transformer-based technologies are currently being deployed in biology, no available agentic system has been developed to tackle bioinformatics workflows. We present a prototype Bioinformatics Retrieval Augmented Data (BRAD) digital assistant. BRAD is a chatbot and agentic system that integrates a suite of tools to handle bioinformatics tasks, from code execution to online search. We demonstrate its capabilities through (1) improved question-and-answering with retrieval augmented generation (RAG), (2) the ability to run complex software pipelines, and (3) the ability to organize and distribute tasks in agentic workflows. We use BRAD for automation, performing tasks ranging from gene enrichment and searching the archive to automatic code generation for running biomarker identification pipelines. BRAD is a step toward autonomous, self-driving labs for digital biology., Comment: 49 pages, 3 tables, 12 figures

This paper analyzes the approximate control variate (ACV) approach to multifidelity uncertainty quantification in the case where weighted estimators are combined to form the components of the ACV. The weighted estimators enable one to precisely group models that share input samples to achieve improved variance reduction. We demonstrate that this viewpoint yields a generalized linear estimator that can assign any weight to any sample. This generalization shows that other linear estimators in the literature, particularly the multilevel best linear unbiased estimator (ML-BLUE) of Schaden and Ullman in 2020, becomes a specific version of the ACV estimator of Gorodetsky, Geraci, Jakeman, and Eldred, 2020. Moreover, this connection enables numerous extensions and insights. For example, we empirically show that having non-independent groups can yield better variance reduction compared to the independent groups used by ML-BLUE. Furthermore, we show that such grouped estimators can use arbitrary weighted estimators, not just the simple Monte Carlo estimators used in ML-BLUE. Furthermore, the analysis enables the derivation of ML-BLUE directly from a variance reduction perspective, rather than a regression perspective., Comment: 17 pages, 3 figures

This paper presents a structure-preserving Bayesian approach for learning nonseparable Hamiltonian systems using stochastic dynamic models allowing for statistically-dependent, vector-valued additive and multiplicative measurement noise. The approach is comprised of three main facets. First, we derive a Gaussian filter for a statistically-dependent, vector-valued, additive and multiplicative noise model that is needed to evaluate the likelihood within the Bayesian posterior. Second, we develop a novel algorithm for cost-effective application of Bayesian system identification to high-dimensional systems. Third, we demonstrate how structure-preserving methods can be incorporated into the proposed framework, using nonseparable Hamiltonians as an illustrative system class. We assess the method's performance based on the forecasting accuracy of a model estimated from single-trajectory data. We compare the Bayesian method to a state-of-the-art machine learning method on a canonical nonseparable Hamiltonian model and a chaotic double pendulum model with small, noisy training datasets. The results show that using the Bayesian posterior as a training objective can yield upwards of 724 times improvement in Hamiltonian mean squared error using training data with up to 10% multiplicative noise compared to a standard training objective. Lastly, we demonstrate the utility of the novel algorithm for parameter estimation of a 64-dimensional model of the spatially-discretized nonlinear Schr\"odinger equation with data corrupted by up to 20% multiplicative noise.