Bayesian optimisation (BO) uses probabilistic surrogate models - usually Gaussian processes (GPs) - for the optimisation of expensive black-box functions. At each BO iteration, the GP hyperparameters are fit to previously-evaluated data by maximising the marginal likelihood. However, this fails to account for uncertainty in the hyperparameters themselves, leading to overconfident model predictions. This uncertainty can be accounted for by taking the Bayesian approach of marginalising out the model hyperparameters. We investigate whether a fully-Bayesian treatment of the Gaussian process hyperparameters in BO (FBBO) leads to improved optimisation performance. Since an analytic approach is intractable, we compare FBBO using three approximate inference schemes to the maximum likelihood approach, using the Expected Improvement (EI) and Upper Confidence Bound (UCB) acquisition functions paired with ARD and isotropic Matern kernels, across 15 well-known benchmark problems for 4 observational noise settings. FBBO using EI with an ARD kernel leads to the best performance in the noise-free setting, with much less difference between combinations of BO components when the noise is increased. FBBO leads to over-exploration with UCB, but is not detrimental with EI. Therefore, we recommend that FBBO using EI with an ARD kernel as the default choice for BO.

We provide quantitative upper bounds on the total variation mixing time of the Markov chain corresponding to the unadjusted Hamiltonian Monte Carlo (uHMC) algorithm. For two general classes of models and fixed time discretization step size $h$, the mixing time is shown to depend only logarithmically on the dimension. Moreover, we provide quantitative upper bounds on the total variation distance between the invariant measure of the uHMC chain and the true target measure. As a consequence, we show that an $\varepsilon$-accurate approximation of the target distribution $\mu$ in total variation distance can be achieved by uHMC for a broad class of models with $O\left(d^{3/4}\varepsilon^{-1/2}\log (d/\varepsilon )\right)$ gradient evaluations, and for mean field models with weak interactions with $O\left(d^{1/2}\varepsilon^{-1/2}\log (d/\varepsilon )\right)$ gradient evaluations. The proofs are based on the construction of successful couplings for uHMC that realize the upper bounds.

The numerical solution of differential equations can be formulated as an inference problem to which formal statistical approaches can be applied. However, nonlinear partial differential equations (PDEs) pose substantial challenges from an inferential perspective, most notably the absence of explicit conditioning formula. This paper extends earlier work on linear PDEs to a general class of initial value problems specified by nonlinear PDEs, motivated by problems for which evaluations of the right-hand-side, initial conditions, or boundary conditions of the PDE have a high computational cost. The proposed method can be viewed as exact Bayesian inference under an approximate likelihood, which is based on discretisation of the nonlinear differential operator. Proof-of-concept experimental results demonstrate that meaningful probabilistic uncertainty quantification for the unknown solution of the PDE can be performed, while controlling the number of times the right-hand-side, initial and boundary conditions are evaluated. A suitable prior model for the solution of the PDE is identified using novel theoretical analysis of the sample path properties of Mat\'{e}rn processes, which may be of independent interest.

We study the problem of minimizing the resource capacity of autonomous agents cooperating to achieve a shared task. More specifically, we consider high-level planning for a team of homogeneous agents that operate under resource constraints in stochastic environments and share a common goal: given a set of target locations, ensure that each location will be visited infinitely often by some agent almost surely. We formalize the dynamics of agents by consumption Markov decision processes. In a consumption Markov decision process, the agent has a resource of limited capacity. Each action of the agent may consume some amount of the resource. To avoid exhaustion, the agent can replenish its resource to full capacity in designated reload states. The resource capacity restricts the capabilities of the agent. The objective is to assign target locations to agents, and each agent is only responsible for visiting the assigned subset of target locations repeatedly. Moreover, the assignment must ensure that the agents can carry out their tasks with minimal resource capacity. We reduce the problem of finding target assignments for a team of agents with the lowest possible capacity to an equivalent graph-theoretical problem. We develop an algorithm that solves this graph problem in time that is \emph{polynomial} in the number of agents, target locations, and size of the consumption Markov decision process. We demonstrate the applicability and scalability of the algorithm in a scenario where hundreds of unmanned underwater vehicles monitor hundreds of locations in environments with stochastic ocean currents.

We propose a new method to estimate causal effects from nonexperimental data. Each pair of sample units is first associated with a stochastic 'treatment' - differences in factors between units - and an effect - a resultant outcome difference. It is then proposed that all such pairs can be combined to provide more accurate estimates of causal effects in observational data, provided a statistical model connecting combinatorial properties of treatments to the accuracy and unbiasedness of their effects. The article introduces one such model and a Bayesian approach to combine the $O(n^2)$ pairwise observations typically available in nonexperimnetal data. This also leads to an interpretation of nonexperimental datasets as incomplete, or noisy, versions of ideal factorial experimental designs. This approach to causal effect estimation has several advantages: (1) it expands the number of observations, converting thousands of individuals into millions of observational treatments; (2) starting with treatments closest to the experimental ideal, it identifies noncausal variables that can be ignored in the future, making estimation easier in each subsequent iteration while departing minimally from experiment-like conditions; (3) it recovers individual causal effects in heterogeneous populations. We evaluate the method in simulations and the National Supported Work (NSW) program, an intensively studied program whose effects are known from randomized field experiments. We demonstrate that the proposed approach recovers causal effects in common NSW samples, as well as in arbitrary subpopulations and an order-of-magnitude larger supersample with the entire national program data, outperforming Statistical, Econometrics and Machine Learning estimators in all cases...

A Bayesian network is a probabilistic graphical model, which represents conditional independence relationships between a set of random variables by a directed acyclic graph (DAG).The problem of DAG learning from observational data is hard (Chickering 1996), and the number of DAGs grows super-exponentially with the number of nodes. Hence, developing and implementing methods to learn an underlying DAG from observational data in reasonable time continues to be the focus of much research (Bartlett and Cussens 2017; Goudie and Mukherjee 2016; Scanagatta, de Campos, and Corani 2015). Drton and Maathuis (2017) provide an overview of the approaches for structure learning of graphical models including Bayesian networks. The R (R Development Core Team 2008) packages pcalg (Kalisch, Mächler, Colombo, Maathuis, and Bühlmann 2012), BNlearn (Scutari 2010), bnstruct (Franzin, Sambo, and Camillo 2017) and the Java-based toolbox TETRAD (Glymour, Scheines, Spirtes, and Ramsey 2017) implement multiple approaches to structure learning, including both constraint-based and searcharXiv:2105.00488v1

The last decade has witnessed an experimental revolution in data science and machine learning, epitomised by deep learning methods. Indeed, many high-dimensional learning tasks previously thought to be beyond reach -- such as computer vision, playing Go, or protein folding -- are in fact feasible with appropriate computational scale. Remarkably, the essence of deep learning is built from two simple algorithmic principles: first, the notion of representation or feature learning, whereby adapted, often hierarchical, features capture the appropriate notion of regularity for each task, and second, learning by local gradient-descent type methods, typically implemented as backpropagation. While learning generic functions in high dimensions is a cursed estimation problem, most tasks of interest are not generic, and come with essential pre-defined regularities arising from the underlying low-dimensionality and structure of the physical world. This text is concerned with exposing these regularities through unified geometric principles that can be applied throughout a wide spectrum of applications. Such a 'geometric unification' endeavour, in the spirit of Felix Klein's Erlangen Program, serves a dual purpose: on one hand, it provides a common mathematical framework to study the most successful neural network architectures, such as CNNs, RNNs, GNNs, and Transformers. On the other hand, it gives a constructive procedure to incorporate prior physical knowledge into neural architectures and provide principled way to build future architectures yet to be invented.

[This paper was initially published in PHME conference in 2016, selected for further publication in International Journal of Prognostics and Health Management.] This paper describes an Autoregressive Partially-hidden Markov model (ARPHMM) for fault detection and prognostics of equipments based on sensors' data. It is a particular dynamic Bayesian network that allows to represent the dynamics of a system by means of a Hidden Markov Model (HMM) and an autoregressive (AR) process. The Markov chain assumes that the system is switching back and forth between internal states while the AR process ensures a temporal coherence on sensor measurements. A sound learning procedure of standard ARHMM based on maximum likelihood allows to iteratively estimate all parameters simultaneously. This paper suggests a modification of the learning procedure considering that one may have prior knowledge about the structure which becomes partially hidden. The integration of the prior is based on the Theory of Weighted Distributions which is compatible with the Expectation-Maximization algorithm in the sense that the convergence properties are still satisfied. We show how to apply this model to estimate the remaining useful life based on health indicators. The autoregressive parameters can indeed be used for prediction while the latent structure can be used to get information about the degradation level. The interest of the proposed method for prognostics and health assessment is demonstrated on CMAPSS datasets.

Bayesian Knowledge Tracing, a model used for cognitive mastery estimation, has been a hallmark of adaptive learning research and an integral component of deployed intelligent tutoring systems (ITS). In this paper, we provide a brief history of knowledge tracing model research and introduce pyBKT, an accessible and computationally efficient library of model extensions from the literature. The library provides data generation, fitting, prediction, and cross-validation routines, as well as a simple to use data helper interface to ingest typical tutor log dataset formats. We evaluate the runtime with various dataset sizes and compare to past implementations. Additionally, we conduct sanity checks of the model using experiments with simulated data to evaluate the accuracy of its EM parameter learning and use real-world data to validate its predictions, comparing pyBKT's supported model variants with results from the papers in which they were originally introduced. The library is open source and open license for the purpose of making knowledge tracing more accessible to communities of research and practice and to facilitate progress in the field through easier replication of past approaches.

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