Large language models are thought to have potential to aid in medical decision making. We investigate this here. We start with the treatment problem, the patient's core medical decision-making task, which is solved in collaboration with a healthcare provider. We discuss approaches to solving the treatment problem, including -- within evidence-based medicine -- trials and observational data. We then discuss the chat problem, and how this differs from the treatment problem -- in particular as it relates to imitation. We then discuss how a large language model might be used to solve the treatment problem and highlight some of the challenges that emerge. We finally discuss how these challenges relate to evidence-based medicine, and how this might inform next steps.

Modern general-purpose artificial intelligence (AI) systems present an urgent risk management challenge, as their rapidly evolving capabilities and potential for catastrophic harm outpace our ability to reliably assess their risks. Current methods often rely on selective testing and undocumented assumptions about risk priorities, frequently failing to make a serious attempt at assessing the set of pathways through which AI systems pose direct or indirect risks to society and the biosphere. This paper introduces the probabilistic risk assessment (PRA) for AI framework, adapting established PRA techniques from high-reliability industries (e.g., nuclear power, aerospace) for the new challenges of advanced AI. The framework guides assessors in identifying potential risks, estimating likelihood and severity bands, and explicitly documenting evidence, underlying assumptions, and analyses at appropriate granularities. The framework's implementation tool synthesizes the results into a risk report card with aggregated risk estimates from all assessed risks. It introduces three methodological advances: (1) Aspect-oriented hazard analysis provides systematic hazard coverage guided by a first-principles taxonomy of AI system aspects (e.g. capabilities, domain knowledge, affordances); (2) Risk pathway modeling analyzes causal chains from system aspects to societal impacts using bidirectional analysis and incorporating prospective techniques; and (3) Uncertainty management employs scenario decomposition, reference scales, and explicit tracing protocols to structure credible projections with novelty or limited data. Additionally, the framework harmonizes diverse assessment methods by integrating evidence into comparable, quantified absolute risk estimates for lifecycle decisions. We have implemented this as a workbook tool for AI developers, evaluators, and regulators.

This paper provides a proof of the consistency of sparse grid quadrature for numerical integration of high dimensional distributions. In a first step, a transport map is learned that normalizes the distribution to a noise distribution on the unit cube. This step is built on the statistical learning theory of neural ordinary differential equations, which has been established recently. Secondly, the composition of the generative map with the quantity of interest is integrated numerically using the Clenshaw-Curtis sparse grid quadrature. A decomposition of the total numerical error in quadrature error and statistical error is provided. As main result it is proven in the framework of empirical risk minimization that all error terms can be controlled in the sense of PAC (probably approximately correct) learning and with high probability the numerical integral approximates the theoretical value up to an arbitrary small error in the limit where the data set size is growing and the network capacity is increased adaptively.

Inferring protein production kinetics for dividing cells is complicated due to protein inheritance from the mother cell. For instance, fluorescence measurements -- commonly used to assess gene activation -- may reflect not only newly produced proteins but also those inherited through successive cell divisions. In such cases, observed protein levels in any given cell are shaped by its division history. As a case study, we examine activation of the glc3 gene in yeast involved in glycogen synthesis and expressed under nutrient-limiting conditions. We monitor this activity using snapshot fluorescence measurements via flow cytometry, where GFP expression reflects glc3 promoter activity. A naïve analysis of flow cytometry data ignoring cell division suggests many cells are active with low expression. Explicitly accounting for the (non-Markovian) effects of cell division and protein inheritance makes it impossible to write down a tractable likelihood -- a key ingredient in physics-inspired inference, defining the probability of observing data given a model. The dependence on a cell's division history breaks the assumptions of standard (Markovian) master equations, rendering traditional likelihood-based approaches inapplicable. Instead, we adapt conditional normalizing flows (a class of neural network models designed to learn probability distributions) to approximate otherwise intractable likelihoods from simulated data. In doing so, we find that glc3 is mostly inactive under stress, showing that while cells occasionally activate the gene, expression is brief and transient.

Gaussian mixture models (GMMs) are ubiquitous in statistical learning, particularly for unsupervised problems. While full GMMs suffer from the overparameterization of their covariance matrices in high-dimensional spaces, spherical GMMs (with isotropic covariance matrices) certainly lack flexibility to fit certain anisotropic distributions. Connecting these two extremes, we introduce a new family of parsimonious GMMs with piecewise-constant covariance eigenvalue profiles. These extend several low-rank models like the celebrated mixtures of probabilistic principal component analyzers (MPPCA), by enabling any possible sequence of eigenvalue multiplicities. If the latter are prespecified, then we can naturally derive an expectation-maximization (EM) algorithm to learn the mixture parameters. Otherwise, to address the notoriously-challenging issue of jointly learning the mixture parameters and hyperparameters, we propose a componentwise penalized EM algorithm, whose monotonicity is proven. We show the superior likelihood-parsimony tradeoffs achieved by our models on a variety of unsupervised experiments: density fitting, clustering and single-image denoising.

This study uses a Variational Autoencoder method to enhance the efficiency and applicability of Markov Chain Monte Carlo (McMC) methods by generating broader-spectrum prior proposals. Traditional approaches, such as the Karhunen-Loève Expansion (KLE), require previous knowledge of the covariance function, often unavailable in practical applications. The VAE framework enables a data-driven approach to flexibly capture a broader range of correlation structures in Bayesian inverse problems, particularly subsurface flow modeling. The methodology is tested on a synthetic groundwater flow inversion problem, where pressure data is used to estimate permeability fields. Numerical experiments demonstrate that the VAE-based parameterization achieves comparable accuracy to KLE when the correlation length is known and outperforms KLE when the assumed correlation length deviates from the true value. Moreover, the VAE approach significantly reduces stochastic dimensionality, improving computational efficiency. The results suggest that leveraging deep generative models in McMC methods can lead to more adaptable and efficient Bayesian inference in high-dimensional problems.

Ordinal classification problems, where labels exhibit a natural order, are prevalent in high-stakes fields such as medicine and finance. Accurate uncertainty quantification, including the decomposition into aleatoric (inherent variability) and epistemic (lack of knowledge) components, is crucial for reliable decision-making. However, existing research has primarily focused on nominal classification and regression. In this paper, we introduce a novel class of measures of aleatoric and epistemic uncertainty in ordinal classification, which is based on a suitable reduction to (entropy- and variance-based) measures for the binary case. These measures effectively capture the trade-off in ordinal classification between exact hit-rate and minimial error distances. We demonstrate the effectiveness of our approach on various tabular ordinal benchmark datasets using ensembles of gradient-boosted trees and multi-layer perceptrons for approximate Bayesian inference. Our method significantly outperforms standard and label-wise entropy and variance-based measures in error detection, as indicated by misclassification rates and mean absolute error. Additionally, the ordinal measures show competitive performance in out-of-distribution (OOD) detection. Our findings highlight the importance of considering the ordinal nature of classification problems when assessing uncertainty.

The idea that Gregorian melodies are constructed from some vocabulary of segments has long been a part of chant scholarship. This so-called "centonisation" theory has received much musicological criticism, but frequent re-use of certain melodic segments has been observed in chant melodies, and the intractable number of possible segmentations allowed the option that some undiscovered segmentation exists that will yet prove the value of centonisation, and recent empirical results have shown that segmentations can outperform music-theoretical features in mode classification. Inspired by the fact that Gregorian chant was memorised, we search for an optimal unsupervised segmentation of chant melody using nested hierarchical Pitman-Yor language models. The segmentation we find achieves state-of-the-art performance in mode classification. Modeling a monk memorising the melodies from one liturgical manuscript, we then find empirical evidence for the link between mode classification and memory efficiency, and observe more formulaic areas at the beginnings and ends of melodies corresponding to the practical role of modality in performance. However, the resulting segmentations themselves indicate that even such a memory-optimal segmentation is not what is understood as centonisation.

We consider a Bayesian diffusion control problem of expected terminal utility maximization. The controller imposes a prior distribution on the unknown drift of an underlying diffusion. The Bayesian optimal control, tracking the posterior distribution of the unknown drift, can be characterized explicitly. However, in practice, the prior will generally be incorrectly specified, and the degree of model misspecification can have a significant impact on policy performance. To mitigate this and reduce overpessimism, we introduce a distributionally robust Bayesian control (DRBC) formulation in which the controller plays a game against an adversary who selects a prior in divergence neighborhood of a baseline prior. The adversarial approach has been studied in economics and efficient algorithms have been proposed in static optimization settings. We develop a strong duality result for our DRBC formulation. Combining these results together with tools from stochastic analysis, we are able to derive a loss that can be efficiently trained (as we demonstrate in our numerical experiments) using a suitable neural network architecture. As a result, we obtain an effective algorithm for computing the DRBC optimal strategy. The methodology for computing the DRBC optimal strategy is greatly simplified, as we show, in the important case in which the adversary chooses a prior from a Kullback-Leibler distributional uncertainty set.

Generative Adversarial Networks (GANs) are popular and successful generative models. Despite their success, optimization is notoriously challenging and they require regularization against overfitting. In this work, we explain the success and limitations of GANs by interpreting them as probabilistic generative models. This interpretation enables us to view GANs as Bayesian neural networks with partial stochasticity, allowing us to establish conditions of universal approximation. We can then cast the adversarial-style optimization of several variants of GANs as the optimization of a proxy for the marginal likelihood. Taking advantage of the connection between marginal likelihood optimization and Occam's razor, we can define regularization and optimization strategies to smooth the loss landscape and search for solutions with minimum description length, which are associated with flat minima and good generalization. The results on a wide range of experiments indicate that these strategies lead to performance improvements and pave the way to a deeper understanding of regularization strategies for GANs.