We introduce Bayesian optimization, a technique developed for optimizing time-consuming engineering simulations and for fitting machine learning models on large datasets. Bayesian optimization guides the choice of experiments during materials design and discovery to find good material designs in as few experiments as possible. We focus on the case when materials designs are parameterized by a low-dimensional vector. Bayesian optimization is built on a statistical technique called Gaussian process regression, which allows predicting the performance of a new design based on previously tested designs. After providing a detailed introduction to Gaussian process regression, we introduce two Bayesian optimization methods: expected improvement, for design problems with noise-free evaluations; and the knowledge-gradient method, which generalizes expected improvement and may be used in design problems with noisy evaluations. Both methods are derived using a value-of-information analysis, and enjoy one-step Bayes-optimality.

We deliver a call to arms for probabilistic numerical methods: algorithms for numerical tasks, including linear algebra, integration, optimization and solving differential equations, that return uncertainties in their calculations. Such uncertainties, arising from the loss of precision induced by numerical calculation with limited time or hardware, are important for much contemporary science and industry. Within applications such as climate science and astrophysics, the need to make decisions on the basis of computations with large and complex data has led to a renewed focus on the management of numerical uncertainty. We describe how several seminal classic numerical methods can be interpreted naturally as probabilistic inference. We then show that the probabilistic view suggests new algorithms that can flexibly be adapted to suit application specifics, while delivering improved empirical performance. We provide concrete illustrations of the benefits of probabilistic numeric algorithms on real scientific problems from astrometry and astronomical imaging, while highlighting open problems with these new algorithms. Finally, we describe how probabilistic numerical methods provide a coherent framework for identifying the uncertainty in calculations performed with a combination of numerical algorithms (e.g. both numerical optimisers and differential equation solvers), potentially allowing the diagnosis (and control) of error sources in computations.

Network metrics form a fundamental part of the network analysis toolbox. Used to quantitatively measure different aspects of the network, these metrics can give insights into the underlying network structure and function. In this work, we connect network metrics to modern probabilistic machine learning. We focus on the centrality metric, which is used a wide variety of applications from web search to gene-analysis. First, we formulate an eigenvector-based Bayesian centrality model for determining node importance. Compared to existing methods, our probabilistic model allows for the assimilation of multiple edge weight observations, the inclusion of priors and the extraction of uncertainties. To enable tractable inference, we develop a variational lower bound (VBC) that is demonstrated to be effective on a variety of networks (two synthetic and five real-world graphs). We then bridge this model to sparse Gaussian processes. The sparse variational Bayesian centrality Gaussian process (VBC-GP) learns a mapping between node attributes to latent centrality and hence, is capable of predicting centralities from node features and can potentially represent a large number of nodes using only a limited number of inducing inputs. Experiments show that the VBC-GP learns high-quality mappings and compares favorably to a two-step baseline, i.e., a full GP trained on the node attributes and pre-computed centralities. Finally, we present two case-studies using the VBC-GP: first, to ascertain relevant features in a taxi transport network and second, to distribute a limited number of vaccines to mitigate the severity of a viral outbreak.

It is well known in the literature that the problem of learning the structure of Bayesian networks is very hard to tackle: its computational complexity is super-exponential in the number of nodes in the worst case and polynomial in most real-world scenarios. Efficient implementations of score-based structure learning benefit from past and current research in optimisation theory, which can be adapted to the task by using the network score as the objective function to maximise. This is not true for approaches based on conditional independence tests, called constraint-based learning algorithms. The only optimisation in widespread use, backtracking, leverages the symmetries implied by the definitions of neighbourhood and Markov blanket. In this paper we illustrate how backtracking is implemented in recent versions of the bnlearn R package, and how it degrades the stability of Bayesian network structure learning for little gain in terms of speed. As an alternative, we describe a software architecture and framework that can be used to parallelise constraint-based structure learning algorithms (also implemented in bnlearn) and we demonstrate its performance using four reference networks and two real-world data sets from genetics and systems biology. We show that on modern multi-core or multiprocessor hardware parallel implementations are preferable over backtracking, which was developed when single-processor machines were the norm.

In data science, it is often required to estimate dependencies between different data sources. These dependencies are typically calculated using Pearson's correlation, distance correlation, and/or mutual information. However, none of these measures satisfy all the Granger's axioms for an "ideal measure". One such ideal measure, proposed by Granger himself, calculates the Bhattacharyya distance between the joint probability density function (pdf) and the product of marginal pdfs. We call this measure the mutual dependence. However, to date this measure has not been directly computable from data. In this paper, we use our recently introduced maximum likelihood non-parametric estimator for band-limited pdfs, to compute the mutual dependence directly from the data. We construct the estimator of mutual dependence and compare its performance to standard measures (Pearson's and distance correlation) for different known pdfs by computing convergence rates, computational complexity, and the ability to capture nonlinear dependencies. Our mutual dependence estimator requires fewer samples to converge to theoretical values, is faster to compute, and captures more complex dependencies than standard measures.

We establish an equivalence between two seemingly different theories: one is the traditional axiomatisation of incomplete preferences on horse lotteries based on the mixture independence axiom; the other is the theory of desirable gambles (bounded random variables) developed in the context of imprecise probability, which we extend here to make it deal with vector-valued gambles. The equivalence allows us to revisit incomplete preferences from the viewpoint, and with the tools, of desirability and through the derived notion of coherent lower previsions (i.e., lower expectation functionals). On this basis, we obtain new results and insights: in particular, we show that the theory of incomplete preferences can be developed assuming only the existence of a worst act--no best act is needed--, and that a weakened Archimedean axiom suffices too; this axiom allows us also to address some controversy about the regularity assumption (that probabilities should be positive--they need not), which enables us also to deal with uncountable possibility spaces; we show that it is always possible to extend in a minimal way a preference relation to one with a worst act, and yet the resulting relation is never Archimedean, except in a trivial case; we show that the traditional notion of state independence coincides with the notion called strong independence in imprecise probability (stochastic independence in the case of complete preferences)--this leads us to give much a weaker definition of state independence than the traditional one; we rework and uniform the notions of complete preferences, beliefs, values; we argue that Archimedeanity does not capture all the problems that can be modelled with sets of expected utilities and we provide a new notion that does precisely that. Perhaps most importantly, we argue throughout that desirability is a powerful and natural setting to model, and work with, incomplete preferences, even in the case of non-Archimedean problems. This leads us to suggest that desirability, rather than preference, should be the primitive notion at the basis of decision-theoretic axiomatisations. Keywords: Incomplete preferences, decision theory, expected utility, desirability, convex cones, imprecise probability.

Models of complex systems are often formalized as sequential software simulators: computationally intensive programs that iteratively build up probable system configurations given parameters and initial conditions. These simulators enable modelers to capture effects that are difficult to characterize analytically or summarize statistically. However, in many real-world applications, these simulations need to be inverted to match the observed data. This typically requires the custom design, derivation and implementation of sophisticated inversion algorithms. Here we give a framework for inverting a broad class of complex software simulators via probabilistic programming and automatic inference, using under 20 lines of probabilistic code. Our approach is based on a formulation of inversion as approximate inference in a simple sequential probabilistic model. We implement four inference strategies, including Metropolis-Hastings, a sequentialized Metropolis-Hastings scheme, and a particle Markov chain Monte Carlo scheme, requiring 4 or fewer lines of probabilistic code each. We demonstrate our framework by applying it to invert a real geological software simulator from the oil and gas industry.

We study the computational complexity of Markov chain Monte Carlo (MCMC) methods for high-dimensional Bayesian linear regression under sparsity constraints. We first show that a Bayesian approach can achieve variable-selection consistency under relatively mild conditions on the design matrix. We then demonstrate that the statistical criterion of posterior concentration need not imply the computational desideratum of rapid mixing of the MCMC algorithm. By introducing a truncated sparsity prior for variable selection, we provide a set of conditions that guarantee both variable-selection consistency and rapid mixing of a particular Metropolis-Hastings algorithm. The mixing time is linear in the number of covariates up to a logarithmic factor. Our proof controls the spectral gap of the Markov chain by constructing a canonical path ensemble that is inspired by the steps taken by greedy algorithms for variable selection.

Stochastic variational inference allows for fast posterior inference in complex Bayesian models. However, the algorithm is prone to local optima which can make the quality of the posterior approximation sensitive to the choice of hyperparameters and initialization. We address this problem by replacing the natural gradient step of stochastic varitional inference with a trust-region update. We show that this leads to generally better results and reduced sensitivity to hyperparameters. We also describe a new strategy for variational inference on streaming data and show that here our trust-region method is crucial for getting good performance.

There has been significant interest in the use of fully-connected graphical models and deep-structured graphical models for the purpose of structured inference. However, fully-connected and deep-structured graphical models have been largely explored independently, leaving the unification of these two concepts ripe for exploration. A fundamental challenge with unifying these two types of models is in dealing with computational complexity. In this study, we investigate the feasibility of unifying fully-connected and deep-structured models in a computationally tractable manner for the purpose of structured inference. To accomplish this, we introduce a deep-structured fully-connected random field (DFRF) model that integrates a series of intermediate sparse auto-encoding layers placed between state layers to significantly reduce computational complexity. The problem of image segmentation was used to illustrate the feasibility of using the DFRF for structured inference in a computationally tractable manner. Results in this study show that it is feasible to unify fully-connected and deep-structured models in a computationally tractable manner for solving structured inference problems such as image segmentation.