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Bayesian probabilistic numerical methods are a set of tools providing posterior distributions on the output of numerical methods. The use of these methods is usually motivated by the fact that they can represent our uncertainty due to…
Bayesian methods - either based on Bayes Factors or BIC - are now widely used for model selection. One property that might reasonably be demanded of any model selection method is that if a model ${M}_{1}$ is preferred to a model ${M}_{0}$,…
Approximate Bayesian computation (ABC) performs statistical inference for otherwise intractable probability models by accepting parameter proposals when corresponding simulated datasets are sufficiently close to the observations. Producing…
This chapter provides a overview of Bayesian inference, mostly emphasising that it is a universal method for summarising uncertainty and making estimates and predictions using probability statements conditional on observed data and an…
Regression for spatially dependent outcomes poses many challenges, for inference and for computation. Non-spatial models and traditional spatial mixed-effects models each have their advantages and disadvantages, making it difficult for…
As an alternative to variable selection or shrinkage in high dimensional regression, we propose to randomly compress the predictors prior to analysis. This dramatically reduces storage and computational bottlenecks, performing well when the…
The ranking problem is to order a collection of units by some unobserved parameter, based on observations from the associated distribution. This problem arises naturally in a number of contexts, such as business, where we may want to rank…
Bayesian methods have been widely used in the last two decades to infer statistical properties of spatially variable coefficients in partial differential equations from measurements of the solutions of these equations. Yet, in many cases…
When the target variable exhibits a semicontinuous behaviour (i.e. a point mass in a single value and a continuous distribution elsewhere) parametric `two-part regression models' have been extensively used and investigated. In this paper, a…
This paper concerns the robust regression model when the number of predictors and the number of observations grow in a similar rate. Theory for M-estimators in this regime has been recently developed by several authors [El Karoui et al.,…
Rich data generating mechanisms are ubiquitous in this age of information and require complex statistical models to draw meaningful inference. While Bayesian analysis has seen enormous development in the last 30 years, benefitting from the…
The emergent field of probabilistic numerics has thus far lacked clear statistical principals. This paper establishes Bayesian probabilistic numerical methods as those which can be cast as solutions to certain inverse problems within the…
Economic evaluations from individual-level data are an important component of the process of technology appraisal, with a view to informing resource allocation decisions. A critical problem in these analyses is that both effectiveness and…
Approximate Bayesian computation (ABC) methods, which are applicable when the likelihood is difficult or impossible to calculate, are an active topic of current research. Most current ABC algorithms directly approximate the posterior…
Binary optimization is a central problem in mathematical optimization and its applications are abundant. To solve this problem, we propose a new class of continuous optimization techniques which is based on Mathematical Programming with…
The Bayesian approach has proved to be a coherent approach to handle ill posed Inverse problems. However, the Bayesian calculations need either an optimization or an integral calculation. The maximum a posteriori (MAP) estimation requires…
Estimating the dependences between random variables, and ranking them accordingly, is a prevalent problem in machine learning. Pursuing frequentist and information-theoretic approaches, we first show that the p-value and the mutual…
This paper introduces and reviews some of the principles and methods used in Bayesian reliability. It specifically discusses methods used in the analysis of success/no-success data and then reminds the reader of a simple Monte Carlo…
Explosive growth in data and availability of cheap computing resources have sparked increasing interest in Big learning, an emerging subfield that studies scalable machine learning algorithms, systems, and applications with Big Data.…
We want to select the best systems out of a given set of systems (or rank them) with respect to their expected performance. The systems allow random observations only and we assume that the joint observation of the systems has a…