Proximal nested sampling was introduced recently to open up Bayesian model selection for high-dimensional problems such as computational imaging. The framework is suitable for models with a log-convex likelihood, which are ubiquitous in the imaging sciences. The purpose of this article is two-fold. First, we review proximal nested sampling in a pedagogical manner in an attempt to elucidate the framework for physical scientists. Second, we show how proximal nested sampling can be extended in an empirical Bayes setting to support data-driven priors, such as deep neural networks learned from training data.

Recent advances of powerful Language Models have allowed Natural Language Generation (NLG) to emerge as an important technology that can not only perform traditional tasks like summarisation or translation, but also serve as a natural language interface to a variety of applications. As such, it is crucial that NLG systems are trustworthy and reliable, for example by indicating when they are likely to be wrong; and supporting multiple views, backgrounds and writing styles -- reflecting diverse human sub-populations. In this paper, we argue that a principled treatment of uncertainty can assist in creating systems and evaluation protocols better aligned with these goals. We first present the fundamental theory, frameworks and vocabulary required to represent uncertainty. We then characterise the main sources of uncertainty in NLG from a linguistic perspective, and propose a two-dimensional taxonomy that is more informative and faithful than the popular aleatoric/epistemic dichotomy. Finally, we move from theory to applications and highlight exciting research directions that exploit uncertainty to power decoding, controllable generation, self-assessment, selective answering, active learning and more.

The famous aphorism of George Box states: "all models are wrong, but some are useful" [1]. It is the task of statisticians and data analysts to find a model which is most useful for a given problem. The build, compute, critique and repeat cycle [2], also known as Box's loop [3], is an iterative approach for finding the most useful model. Any efforts in shortening this design cycle increase the chances of developing more useful models, which in turn might yield more reliable predictions, more profitable returns or more efficient operations for the problem at hand. In this paper we choose to adopt the Bayesian formalism and therefore we will specify all tasks in Box's loop as principled probabilistic inference tasks. In addition to the well-known parameter and state inference tasks, the critique step in the design cycle is also phrased as an inference task, known as Bayesian model comparison, which automatically embodies Occam's razor [4, Ch. 28.1]. Opposed to just selecting a single model in the critique step, for different models we better quantify our confidence about which model is best, especially when data is limited [5, Ch. 18.5.1]. The uncertainty arising from prior beliefs p(m) over a set of models m and limited observations can be naturally included through the use of Bayes' theorem

This paper proposes a novel framework for certifying the fairness of predictive models trained on biased data. It draws from query answering for incomplete and inconsistent databases to formulate the problem of consistent range approximation (CRA) of fairness queries for a predictive model on a target population. The framework employs background knowledge of the data collection process and biased data, working with or without limited statistics about the target population, to compute a range of answers for fairness queries. Using CRA, the framework builds predictive models that are certifiably fair on the target population, regardless of the availability of external data during training. The framework's efficacy is demonstrated through evaluations on real data, showing substantial improvement over existing state-of-the-art methods.

This paper investigates the problem of efficient constrained global optimization of hybrid models that are a composition of a known white-box function and an expensive multi-output black-box function subject to noisy observations, which often arises in real-world science and engineering applications. We propose a novel method, Constrained Upper Quantile Bound (CUQB), to solve such problems that directly exploits the composite structure of the objective and constraint functions that we show leads substantially improved sampling efficiency. CUQB is a conceptually simple, deterministic approach that avoid constraint approximations used by previous methods. Although the CUQB acquisition function is not available in closed form, we propose a novel differentiable sample average approximation that enables it to be efficiently maximized. We further derive bounds on the cumulative regret and constraint violation under a non-parametric Bayesian representation of the black-box function. Since these bounds depend sublinearly on the number of iterations under some regularity assumptions, we establis bounds on the convergence rate to the optimal solution of the original constrained problem. In contrast to most existing methods, CUQB further incorporates a simple infeasibility detection scheme, which we prove triggers in a finite number of iterations when the original problem is infeasible (with high probability given the Bayesian model). Numerical experiments on several test problems, including environmental model calibration and real-time optimization of a reactor system, show that CUQB significantly outperforms traditional Bayesian optimization in both constrained and unconstrained cases. Furthermore, compared to other state-of-the-art methods that exploit composite structure, CUQB achieves competitive empirical performance while also providing substantially improved theoretical guarantees.

--As the network scale increases, existing fully distributed solutions start to lag behind the real-world challenges such as (1) slow information propagation, (2) network communication failures, and (3) external adversarial attacks. In this paper, we focus on hierarchical system architecture and address the problem of non-Bayesian learning over networks that are vulnerable to communication failures and adversarial attacks. On network communication, we consider packet-dropping link failures. We first propose a hierarchical robust push-sum algorithm that can achieve average consensus despite frequent packet-dropping link failures. We provide a sparse information fusion rule between the parameter server and arbitrarily selected network representatives. Then, interleaving the consensus update step with a dual averaging update with Kullback-Leibler (KL) divergence as the proximal function, we obtain a packet-dropping fault-tolerant non-Bayesian learning algorithm with provable convergence guarantees. On external adversarial attacks, we consider Byzantine attacks in which the compromised agents can send maliciously calibrated messages to others (including both the agents and the parameter server). T o avoid the curse of dimensionality of Byzantine consensus, we solve the non-Bayesian learning problem via running multiple dynamics, each of which only involves Byzantine consensus with scalar inputs. T o facilitate resilient information propagation across sub-networks, we use a novel Byzantine-resilient gossiping-type rule at the parameter server . As the scale of the multi-agent network increases, existing fully distributed solutions start to lag behind the crucial real-world challenges such as (1) slow information propagation, (2) network communication failures, and (3) external adversarial attacks.

AB testing aids business operators with their decision making, and is considered the gold standard method for learning from data to improve digital user experiences. However, there is usually a gap between the requirements of practitioners, and the constraints imposed by the statistical hypothesis testing methodologies commonly used for analysis of AB tests. These include the lack of statistical power in multivariate designs with many factors, correlations between these factors, the need of sequential testing for early stopping, and the inability to pool knowledge from past tests. Here, we propose a solution that applies hierarchical Bayesian estimation to address the above limitations. In comparison to current sequential AB testing methodology, we increase statistical power by exploiting correlations between factors, enabling sequential testing and progressive early stopping, without incurring excessive false positive risk. We also demonstrate how this methodology can be extended to enable the extraction of composite global learnings from past AB tests, to accelerate future tests. We underpin our work with a solid theoretical framework that articulates the value of hierarchical estimation. We demonstrate its utility using both numerical simulations and a large set of real-world AB tests. Together, these results highlight the practical value of our approach for statistical inference in the technology industry.

Neural network (NN) potentials promise highly accurate molecular dynamics (MD) simulations within the computational complexity of classical MD force fields. However, when applied outside their training domain, NN potential predictions can be inaccurate, increasing the need for Uncertainty Quantification (UQ). Bayesian modeling provides the mathematical framework for UQ, but classical Bayesian methods based on Markov chain Monte Carlo (MCMC) are computationally intractable for NN potentials. By training graph NN potentials for coarse-grained systems of liquid water and alanine dipeptide, we demonstrate here that scalable Bayesian UQ via stochastic gradient MCMC (SG-MCMC) yields reliable uncertainty estimates for MD observables. We show that cold posteriors can reduce the required training data size and that for reliable UQ, multiple Markov chains are needed. Additionally, we find that SG-MCMC and the Deep Ensemble method achieve comparable results, despite shorter training and less hyperparameter tuning of the latter. We show that both methods can capture aleatoric and epistemic uncertainty reliably, but not systematic uncertainty, which needs to be minimized by adequate modeling to obtain accurate credible intervals for MD observables. Our results represent a step towards accurate UQ that is of vital importance for trustworthy NN potential-based MD simulations required for decision-making in practice.

We study reinforcement learning (RL) for learning a Quantal Stackelberg Equilibrium (QSE) in an episodic Markov game with a leader-follower structure. In specific, at the outset of the game, the leader announces her policy to the follower and commits to it. The follower observes the leader's policy and, in turn, adopts a quantal response policy by solving an entropy-regularized policy optimization problem induced by leader's policy. The goal of the leader is to find her optimal policy, which yields the optimal expected total return, by interacting with the follower and learning from data. A key challenge of this problem is that the leader cannot observe the follower's reward, and needs to infer the follower's quantal response model from his actions against leader's policies. We propose sample-efficient algorithms for both the online and offline settings, in the context of function approximation. Our algorithms are based on (i) learning the quantal response model via maximum likelihood estimation and (ii) model-free or model-based RL for solving the leader's decision making problem, and we show that they achieve sublinear regret upper bounds. Moreover, we quantify the uncertainty of these estimators and leverage the uncertainty to implement optimistic and pessimistic algorithms for online and offline settings. Besides, when specialized to the linear and myopic setting, our algorithms are also computationally efficient. Our theoretical analysis features a novel performance-difference lemma which incorporates the error of quantal response model, which might be of independent interest.

One of the main tasks of actuaries and data scientists is to build good predictive models for certain phenomena such as the claim size or the number of claims in insurance. These models ideally exploit given feature information to enhance the accuracy of prediction. This user guide revisits and clarifies statistical techniques to assess the calibration or adequacy of a model on the one hand, and to compare and rank different models on the other hand. In doing so, it emphasises the importance of specifying the prediction target functional at hand a priori (e.g. the mean or a quantile) and of choosing the scoring function in model comparison in line with this target functional. Guidance for the practical choice of the scoring function is provided. Striving to bridge the gap between science and daily practice in application, it focuses mainly on the pedagogical presentation of existing results and of best practice. The results are accompanied and illustrated by two real data case studies on workers' compensation and customer churn.