While the Bayesian decision-theoretic framework offers an elegant solution to the problem of decision making under uncertainty, one question is how to appropriately select the prior distribution. One idea is to employ a worst-case prior. However, this is not as easy to specify in sequential decision making as in simple statistical estimation problems. This paper studies (sometimes approximate) minimax-Bayes solutions for various reinforcement learning problems to gain insights into the properties of the corresponding priors and policies. We find that while the worst-case prior depends on the setting, the corresponding minimax policies are more robust than those that assume a standard (i.e. uniform) prior.

In this article, we propose a novel pessimism-based Bayesian learning method for optimal dynamic treatment regimes in the offline setting. When the coverage condition does not hold, which is common for offline data, the existing solutions would produce sub-optimal policies. The pessimism principle addresses this issue by discouraging recommendation of actions that are less explored conditioning on the state. However, nearly all pessimism-based methods rely on a key hyper-parameter that quantifies the degree of pessimism, and the performance of the methods can be highly sensitive to the choice of this parameter. We propose to integrate the pessimism principle with Thompson sampling and Bayesian machine learning for optimizing the degree of pessimism. We derive a credible set whose boundary uniformly lower bounds the optimal Q-function, and thus we do not require additional tuning of the degree of pessimism. We develop a general Bayesian learning method that works with a range of models, from Bayesian linear basis model to Bayesian neural network model. We develop the computational algorithm based on variational inference, which is highly efficient and scalable. We establish the theoretical guarantees of the proposed method, and show empirically that it outperforms the existing state-of-the-art solutions through both simulations and a real data example.

Over the past three decades, the logic programming paradigm has been successfully expanded to support probabilistic modeling, inference and learning. The resulting paradigm of probabilistic logic programming (PLP) and its programming languages owes much of its success to a declarative semantics, the so-called distribution semantics. However, the distribution semantics is limited to discrete random variables only. While PLP has been extended in various ways for supporting hybrid, that is, mixed discrete and continuous random variables, we are still lacking a declarative semantics for hybrid PLP that not only generalizes the distribution semantics and the modeling language but also the standard inference algorithm that is based on knowledge compilation. We contribute the hybrid distribution semantics together with the hybrid PLP language DC-ProbLog and its inference engine infinitesimal algebraic likelihood weighting (IALW). These have the original distribution semantics, standard PLP languages such as ProbLog, and standard inference engines for PLP based on knowledge compilation as special cases. Thus, we generalize the state-of-the-art of PLP towards hybrid PLP in three different aspects: semantics, language and inference. Furthermore, IALW is the first inference algorithm for hybrid probabilistic programming based on knowledge compilation.

In fields such as medicine and drug discovery, the ultimate goal of a classification is not to guess a class, but to choose the optimal course of action among a set of possible ones, usually not in one-one correspondence with the set of classes. This decision-theoretic problem requires sensible probabilities for the classes. Probabilities conditional on the features are computationally almost impossible to find in many important cases. The main idea of the present work is to calculate probabilities conditional not on the features, but on the trained classifier's output. This calculation is cheap, needs to be made only once, and provides an output-to-probability "transducer" that can be applied to all future outputs of the classifier. In conjunction with problem-dependent utilities, the probabilities of the transducer allow us to find the optimal choice among the classes or among a set of more general decisions, by means of expected-utility maximization. This idea is demonstrated in a simplified drug-discovery problem with a highly imbalanced dataset. The transducer and utility maximization together always lead to improved results, sometimes close to theoretical maximum, for all sets of problem-dependent utilities. The one-time-only calculation of the transducer also provides, automatically: (i) a quantification of the uncertainty about the transducer itself; (ii) the expected utility of the augmented algorithm (including its uncertainty), which can be used for algorithm selection; (iii) the possibility of using the algorithm in a "generative mode", useful if the training dataset is biased.

Bayesian model comparison (BMC) offers a principled probabilistic approach to study and rank competing models. In standard BMC, we construct a discrete probability distribution over the set of possible models, conditional on the observed data of interest. These posterior model probabilities (PMPs) are measures of uncertainty, but -- when derived from a finite number of observations -- are also uncertain themselves. In this paper, we conceptualize distinct levels of uncertainty which arise in BMC. We explore a fully probabilistic framework for quantifying meta-uncertainty, resulting in an applied method to enhance any BMC workflow. Drawing on both Bayesian and frequentist techniques, we represent the uncertainty over the uncertain PMPs via meta-models which combine simulated and observed data into a predictive distribution for PMPs on new data. We demonstrate the utility of the proposed method in the context of conjugate Bayesian regression, likelihood-based inference with Markov chain Monte Carlo, and simulation-based inference with neural networks.

In this paper, we consider the task of clustering a set of individual time series while modeling each cluster, that is, model-based time series clustering. The task requires a parametric model with sufficient flexibility to describe the dynamics in various time series. To address this problem, we propose a novel model-based time series clustering method with mixtures of linear Gaussian state space models, which have high flexibility. The proposed method uses a new expectation-maximization algorithm for the mixture model to estimate the model parameters, and determines the number of clusters using the Bayesian information criterion. Experiments on a simulated dataset demonstrate the effectiveness of the method in clustering, parameter estimation, and model selection. The method is applied to real datasets commonly used to evaluate time series clustering methods. Results showed that the proposed method produces clustering results that are as accurate or more accurate than those obtained using previous methods.

Unsupervised learning with functional data is an emerging paradigm of machine learning research with applications to computer vision, climate modeling and physical systems. A natural way of modeling functional data is by learning operators between infinite dimensional spaces, leading to discretization invariant representations that scale independently of the sample grid resolution. Here we present Variational Autoencoding Neural Operators (VANO), a general strategy for making a large class of operator learning architectures act as variational autoencoders. For this purpose, we provide a novel rigorous mathematical formulation of the variational objective in function spaces for training. VANO first maps an input function to a distribution over a latent space using a parametric encoder and then decodes a sample from the latent distribution to reconstruct the input, as in classic variational autoencoders. We test VANO with different model set-ups and architecture choices for a variety of benchmarks. We start from a simple Gaussian random field where we can analytically track what the model learns and progressively transition to more challenging benchmarks including modeling phase separation in Cahn-Hilliard systems and real world satellite data for measuring Earth surface deformation.

Determining the most informative features for predicting the overall survival of patients diagnosed with high-grade gastroenteropancreatic neuroendocrine neoplasms is crucial to improve individual treatment plans for patients, as well as the biological understanding of the disease. Recently developed ensemble feature selectors like the Repeated Elastic Net Technique for Feature Selection (RENT) and the User-Guided Bayesian Framework for Feature Selection (UBayFS) allow the user to identify such features in datasets with low sample sizes. While RENT is purely data-driven, UBayFS is capable of integrating expert knowledge a priori in the feature selection process. In this work we compare both feature selectors on a dataset comprising of 63 patients and 134 features from multiple sources, including basic patient characteristics, baseline blood values, tumor histology, imaging, and treatment information. Our experiments involve data-driven and expert-driven setups, as well as combinations of both. We use findings from clinical literature as a source of expert knowledge. Our results demonstrate that both feature selectors allow accurate predictions, and that expert knowledge has a stabilizing effect on the feature set, while the impact on predictive performance is limited. The features WHO Performance Status, Albumin, Platelets, Ki-67, Tumor Morphology, Total MTV, Total TLG, and SUVmax are the most stable and predictive features in our study.

Neuroscientific studies aim to find an accurate and reliable brain Effective Connectome (EC). Although current EC discovery methods have contributed to our understanding of brain organization, their performances are severely constrained by the short sample size and poor temporal resolution of fMRI data, and high dimensionality of the brain connectome. By leveraging the DTI data as prior knowledge, we introduce two Bayesian causal discovery frameworks -- the Bayesian GOLEM (BGOLEM) and Bayesian FGES (BFGES) methods -- that offer significantly more accurate and reliable ECs and address the shortcomings of the existing causal discovery methods in discovering ECs based on only fMRI data. Through a series of simulation studies on synthetic and hybrid (DTI of the Human Connectome Project (HCP) subjects and synthetic fMRI) data, we demonstrate the effectiveness of the proposed methods in discovering EC. To numerically assess the improvement in the accuracy of ECs with our method on empirical data, we first introduce the Pseudo False Discovery Rate (PFDR) as a new computational accuracy metric for causal discovery in the brain. We show that our Bayesian methods achieve higher accuracy than traditional methods on HCP data. Additionally, we measure the reliability of discovered ECs using the Rogers-Tanimoto index for test-retest data and show that our Bayesian methods provide significantly more reproducible ECs than traditional methods. Overall, our study's numerical and graphical results highlight the potential for these frameworks to advance our understanding of brain function and organization significantly.

We consider the general problem of recovering a high-dimensional signal from noisy quantized measurements. Quantization, especially coarse quantization such as 1-bit sign measurements, leads to severe information loss and thus a good prior knowledge of the unknown signal is helpful for accurate recovery. Motivated by the power of score-based generative models (SGM, also known as diffusion models) in capturing the rich structure of natural signals beyond simple sparsity, we propose an unsupervised data-driven approach called quantized compressed sensing with SGM (QCS-SGM), where the prior distribution is modeled by a pre-trained SGM. To perform posterior sampling, an annealed pseudo-likelihood score called noise perturbed pseudo-likelihood score is introduced and combined with the prior score of SGM. The proposed QCS-SGM applies to an arbitrary number of quantization bits. Experiments on a variety of baseline datasets demonstrate that the proposed QCS-SGM significantly outperforms existing state-of-the-art algorithms by a large margin for both in-distribution and out-of-distribution samples. Moreover, as a posterior sampling method, QCS-SGM can be easily used to obtain confidence intervals or uncertainty estimates of the reconstructed results. The code is available at https://github.com/mengxiangming/QCS-SGM.