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• JOURNAL ARTICLE
Ellam L, Girolami M, Pavliotis GA, Wilson Aet al., 2018,

### Stochastic modelling of urban structure

, PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES, Vol: 474, ISSN: 1364-5021
• JOURNAL ARTICLE
Barp A, Briol F-X, Kennedy AD, Girolami Met al., 2018,

### Geometry and dynamics for Markov chain Monte Carlo

, Annual Review of Statistics and Its Application, Vol: 5, Pages: 451-471, ISSN: 2326-8298

Markov Chain Monte Carlo methods have revolutionised mathematical computationand enabled statistical inference within many previously intractable models. Inthis context, Hamiltonian dynamics have been proposed as an efficient way ofbuilding chains which can explore probability densities efficiently. The methodemerges from physics and geometry and these links have been extensively studiedby a series of authors through the last thirty years. However, there iscurrently a gap between the intuitions and knowledge of users of themethodology and our deep understanding of these theoretical foundations. Theaim of this review is to provide a comprehensive introduction to the geometrictools used in Hamiltonian Monte Carlo at a level accessible to statisticians,machine learners and other users of the methodology with only a basicunderstanding of Monte Carlo methods. This will be complemented with somediscussion of the most recent advances in the field which we believe willbecome increasingly relevant to applied scientists.

• JOURNAL ARTICLE
Ellam L, Strathmann H, Girolami M, Murray Iet al., 2017,

### A determinant-free method to simulate the parameters of large Gaussian fields

, STAT, Vol: 6, Pages: 271-281, ISSN: 2049-1573
• JOURNAL ARTICLE
Chkrebtii OA, Campbell DA, Calderhead B, Girolami MAet al., 2016,

### Bayesian Solution Uncertainty Quantification for Differential Equations

, BAYESIAN ANALYSIS, Vol: 11, Pages: 1239-1267, ISSN: 1931-6690
• JOURNAL ARTICLE
Chkrebtii OA, Campbell DA, Calderhead B, Girolami MAet al., 2016,

### o Rejoinder

, BAYESIAN ANALYSIS, Vol: 11, Pages: 1295-1299, ISSN: 1931-6690
• JOURNAL ARTICLE
Ellam L, Zabaras N, Girolami M, 2016,

### A Bayesian approach to multiscale inverse problems with on-the-fly scale determination

, Journal of Computational Physics, Vol: 326, Pages: 115-140, ISSN: 0021-9991

© 2016 Elsevier Inc. A Bayesian computational approach is presented to provide a multi-resolution estimate of an unknown spatially varying parameter from indirect measurement data. In particular, we are interested in spatially varying parameters with multiscale characteristics. In our work, we consider the challenge of not knowing the characteristic length scale(s) of the unknown a priori, and present an algorithm for on-the-fly scale determination. Our approach is based on representing the spatial field with a wavelet expansion. Wavelet basis functions are hierarchically structured, localized in both spatial and frequency domains and tend to provide sparse representations in that a large number of wavelet coefficients are approximately zero. For these reasons, wavelet bases are suitable for representing permeability fields with non-trivial correlation structures. Moreover, the intra-scale correlations between wavelet coefficients form a quadtree, and this structure is exploited to identify additional basis functions to refine the model. Bayesian inference is performed using a sequential Monte Carlo (SMC) sampler with a Markov Chain Monte Carlo (MCMC) transition kernel. The SMC sampler is used to move between posterior densities defined on different scales, thereby providing a computationally efficient method for adaptive refinement of the wavelet representation. We gain insight from the marginal likelihoods, by computing Bayes factors, for model comparison and model selection. The marginal likelihoods provide a termination criterion for our scale determination algorithm. The Bayesian computational approach is rather general and applicable to several inverse problems concerning the estimation of a spatially varying parameter. The approach is demonstrated with permeability estimation for groundwater flow using pressure sensor measurements.

• JOURNAL ARTICLE
Briol F-X, Cockayne J, Teymur O, 2016,

### Contributed Discussion on Article by Chkrebtii, Campbell, Calderhead, and Girolami

, Bayesian Analysis, Vol: 11, Pages: 1285-1293, ISSN: 1931-6690

We commend the authors for an exciting paper which provides a strongcontribution to the emerging field of probabilistic numerics (PN). Below, we discuss aspects of prior modelling which need to be considered thoroughly in future work

• JOURNAL ARTICLE
Epstein M, Calderhead B, Girolami MA, Sivilotti LGet al., 2016,

### Bayesian Statistical Inference in Ion-Channel Models with Exact Missed Event Correction

, BIOPHYSICAL JOURNAL, Vol: 111, Pages: 333-348, ISSN: 0006-3495
• CONFERENCE PAPER
Briol FX, Oates CJ, Girolami M, Osborne MAet al., 2015,

### Frank-Wolfe Bayesian quadrature: Probabilistic integration with theoretical guarantees

, Pages: 1162-1170, ISSN: 1049-5258

Probabilistic integration formulates integration as a statistical inference problem, and is motivated by obtaining a full distribution over numerical error that can be propagated through subsequent computation. Current methods, such as Bayesian Quadrature, demonstrate impressive empirical performance but lack theoretical analysis. An important challenge is therefore to reconcile these methods with rigorous convergence guarantees. In this paper, we present the first probabilistic integrator that admits such theoretical treatment, called Frank-Wolfe Bayesian Quadrature (FWBQ). Under FWBQ, convergence to the true value of the integral is shown to be up to exponential and posterior contraction rates are proven to be up to super-exponential. In simulations, FWBQ is competitive with state-of-the-art methods and out-performs alternatives based on Frank-Wolfe optimisation. Our approach is applied to successfully quantify numerical error in the solution to a challenging Bayesian model choice problem in cellular biology.

• JOURNAL ARTICLE
Girolami MA, 2014,

### Big Bayes Stories: A Collection of Vignettes

, STATISTICAL SCIENCE, Vol: 29, Pages: 97-97, ISSN: 0883-4237
• JOURNAL ARTICLE
Stathopoulos V, Girolami MA, 2013,

### Markov chain Monte Carlo inference for Markov jump processes via the linear noise approximation

, PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES, Vol: 371, ISSN: 1364-503X
• CONFERENCE PAPER
Oates CJ, Niederer S, Lee A, Briol F-X, Girolami Met al.,

### Probabilistic Models for Integration Error in the Assessment of Functional Cardiac Models

This paper studies the numerical computation of integrals, representingestimates or predictions, over the output $f(x)$ of a computational model withrespect to a distribution $p(\mathrm{d}x)$ over uncertain inputs $x$ to themodel. For the functional cardiac models that motivate this work, neither $f$nor $p$ possess a closed-form expression and evaluation of either requires$\approx$ 100 CPU hours, precluding standard numerical integration methods. Ourproposal is to treat integration as an estimation problem, with a joint modelfor both the a priori unknown function $f$ and the a priori unknowndistribution $p$. The result is a posterior distribution over the integral thatexplicitly accounts for dual sources of numerical approximation error due to aseverely limited computational budget. This construction is applied to account,in a statistically principled manner, for the impact of numerical errors that(at present) are confounding factors in functional cardiac model assessment.

• JOURNAL ARTICLE
Briol F-X, Oates CJ, Girolami M, Osborne MA, Sejdinovic Det al.,

### Probabilistic Integration: A Role in Statistical Computation?

A research frontier has emerged in scientific computation, wherein numericalerror is regarded as a source of epistemic uncertainty that can be modelled.This raises several statistical challenges, including the design of statisticalmethods that enable the coherent propagation of probabilities through a(possibly deterministic) computational work-flow. This paper examines the casefor probabilistic numerical methods in routine statistical computation. Ourfocus is on numerical integration, where a probabilistic integrator is equippedwith a full distribution over its output that reflects the presence of anunknown numerical error. Our main technical contribution is to establish, forthe first time, rates of posterior contraction for these methods. These showthat probabilistic integrators can in principle enjoy the "best of bothworlds", leveraging the sampling efficiency of Monte Carlo methods whilstproviding a principled route to assess the impact of numerical error onscientific conclusions. Several substantial applications are provided forillustration and critical evaluation, including examples from statisticalmodelling, computer graphics and a computer model for an oil reservoir.

• CONFERENCE PAPER
Briol F-X, Oates CJ, Cockayne J, Chen WY, Girolami Met al.,

### On the Sampling Problem for Kernel Quadrature

, International Conference on Machine Learning (ICML), Publisher: PMLR, Pages: 586-595

The standard Kernel Quadrature method for numerical integration with random point sets (also called Bayesian Monte Carlo) is known to converge in root mean square error at a rate determined by the ratio $s/d$, where $s$ and $d$ encode the smoothness and dimension of the integrand. However, an empirical investigation reveals that the rate constant $C$ is highly sensitive to the distribution of the random points. In contrast to standard Monte Carlo integration, for which optimal importance sampling is well-understood, the sampling distribution that minimises $C$ for Kernel Quadrature does not admit a closed form. This paper argues that the practical choice of sampling distribution is an important open problem. One solution is considered; a novel automatic approach based on adaptive tempering and sequential Monte Carlo.Empirical results demonstrate a dramatic reduction in integration error of up to 4 orders of magnitude can be achieved with the proposed method.

• JOURNAL ARTICLE
Oates CJ, Cockayne J, Briol F-X, Girolami Met al.,

### Convergence Rates for a Class of Estimators Based on Stein's Method

Gradient information on the sampling distribution can be used to reduce thevariance of Monte Carlo estimators via Stein's method. An important applicationis that of estimating an expectation of a test function along the sample pathof a Markov chain, where gradient information enables convergence rateimprovement at the cost of a linear system which must be solved. Thecontribution of this paper is to establish theoretical bounds on convergencerates for a class of estimators based on Stein's method. Our analysis accountsfor (i) the degree of smoothness of the sampling distribution and testfunction, (ii) the dimension of the state space, and (iii) the case ofnon-independent samples arising from a Markov chain. These results provideinsight into the rapid convergence of gradient-based estimators observed forlow-dimensional problems, as well as clarifying a curse-of-dimension thatappears inherent to such methods.

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