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Chris. J. Oates

Publications and source records attributed to Chris. J. Oates.

46 records · Page 3Linked to original sources
arXiv2018

Posterior Integration on a Riemannian Manifold

The geodesic Markov chain Monte Carlo method and its variants enable computation of integrals with respect to a posterior supported on a manifold. However, for regular integrals, the convergence rate of the ergodic average will be sub-optimal. To fill this gap, this paper extends the efficient posterior integration method of Oates et al. (2017) to the case of a Riemannian manifold. In contrast to the original Euclidean case, no non-trivial boundary conditions are needed for a closed manifold. The method is assessed through simulation and deployed to compute posterior integrals for an Australian Mesozoic paleomagnetic pole model, whose parameters are constrained to lie on the manifold $M = \mathbb{S}^2 \times \mathbb{R}_+$.

stat.ME↗
arXiv2018

Optimal Monte Carlo integration on closed manifolds

The worst case integration error in reproducing kernel Hilbert spaces of standard Monte Carlo methods with n random points decays as $n^{-1/2}$. However, re-weighting of random points can sometimes be used to improve the convergence order. This paper contributes general theoretical results for Sobolev spaces on closed Riemannian manifolds, where we verify that such re-weighting yields optimal approximation rates up to a logarithmic factor. We also provide numerical experiments matching the theoretical results for some Sobolev spaces on the unit sphere and on the Grassmannian manifold. Our theoretical findings also cover function spaces on more general sets such as the unit ball, the cube, and the simplex.

math.NA↗
arXiv2017

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

This paper studies the numerical computation of integrals, representing estimates or predictions, over the output $f(x)$ of a computational model with respect to a distribution $p(\mathrm{d}x)$ over uncertain inputs $x$ to the model. 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. Our proposal is to treat integration as an estimation problem, with a joint model for both the a priori unknown function $f$ and the a priori unknown distribution $p$. The result is a posterior distribution over the integral that explicitly accounts for dual sources of numerical approximation error due to a severely 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.

stat.ME↗
arXiv2017

Probabilistic Integration: A Role in Statistical Computation?

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

stat.ML↗
arXiv2016

Control Functionals for Quasi-Monte Carlo Integration

Quasi-Monte Carlo (QMC) methods are being adopted in statistical applications due to the increasingly challenging nature of numerical integrals that are now routinely encountered. For integrands with $d$-dimensions and derivatives of order $α$, an optimal QMC rule converges at a best-possible rate $O(N^{-α/d})$. However, in applications the value of $α$ can be unknown and/or a rate-optimal QMC rule can be unavailable. Standard practice is to employ $α_L$-optimal QMC where the lower bound $α_L \leq α$ is known, but in general this does not exploit the full power of QMC. One solution is to trade-off numerical integration with functional approximation. This strategy is explored herein and shown to be well-suited to modern statistical computation. A challenging application to robotic arm data demonstrates a substantial variance reduction in predictions for mechanical torques.

stat.CO↗
arXiv2015

Accelerated Nonparametrics for Cascades of Poisson Processes

Cascades of Poisson processes are probabilistic models for spatio-temporal phenomena in which (i) previous events may trigger subsequent events, and (ii) both the background and triggering processes are conditionally Poisson. Such phenomena are typically "data rich but knowledge poor", in the sense that large datasets are available yet a mechanistic understanding of the background and triggering processes which generate the data are unavailable. In these settings nonparametric estimation plays a central role. However existing nonparametric estimators have computational and storage complexity $\mathcal{O}(N^2)$, precluding their application on large datasets. Here, by assuming the triggering process acts only locally, we derive nonparametric estimators with computational complexity $\mathcal{O}(N\log N)$ and storage complexity $\mathcal{O}(N)$. Our approach automatically learns the domain of the triggering process from data and is essentially free from hyperparameters. The methodology is applied to a large seismic dataset where estimation under existing algorithms would be infeasible.

stat.AP↗
arXiv2015

Exploiting Multi-Core Architectures for Reduced-Variance Estimation with Intractable Likelihoods

Many popular statistical models for complex phenomena are intractable, in the sense that the likelihood function cannot easily be evaluated. Bayesian estimation in this setting remains challenging, with a lack of computational methodology to fully exploit modern processing capabilities. In this paper we introduce novel control variates for intractable likelihoods that can dramatically reduce the Monte Carlo variance of Bayesian estimators. We prove that our control variates are well-defined and provide a positive variance reduction. Furthermore we show how to optimise these control variates for variance reduction. The methodology is highly parallel and offers a route to exploit multi-core processing architectures that complements recent research in this direction. Indeed, our work shows that it may not be necessary to parallelise the sampling process itself in order to harness the potential of massively multi-core architectures. Simulation results presented on the Ising model, exponential random graph models and non-linear stochastic differential equation models support our theoretical findings.

stat.CO↗
arXiv2012

Network inference and biological dynamics

Network inference approaches are now widely used in biological applications to probe regulatory relationships between molecular components such as genes or proteins. Many methods have been proposed for this setting, but the connections and differences between their statistical formulations have received less attention. In this paper, we show how a broad class of statistical network inference methods, including a number of existing approaches, can be described in terms of variable selection for the linear model. This reveals some subtle but important differences between the methods, including the treatment of time intervals in discretely observed data. In developing a general formulation, we also explore the relationship between single-cell stochastic dynamics and network inference on averages over cells. This clarifies the link between biochemical networks as they operate at the cellular level and network inference as carried out on data that are averages over populations of cells. We present empirical results, comparing thirty-two network inference methods that are instances of the general formulation we describe, using two published dynamical models. Our investigation sheds light on the applicability and limitations of network inference and provides guidance for practitioners and suggestions for experimental design.

stat.AP↗
arXiv2012

On the relationship between ODEs and DBNs

Recently, Li et al. (Bioinformatics 27(19), 2686-91, 2011) proposed a method, called Differential Equation-based Local Dynamic Bayesian Network (DELDBN), for reverse engineering gene regulatory networks from time-course data. We commend the authors for an interesting paper that draws attention to the close relationship between dynamic Bayesian networks (DBNs) and differential equations (DEs). Their central claim is that modifying a DBN to model Euler approximations to the gradient rather than expression levels themselves is beneficial for network inference. The empirical evidence provided is based on time-course data with equally-spaced observations. However, as we discuss below, in the particular case of equally-spaced observations, Euler approximations and conventional DBNs lead to equivalent statistical models that, absent artefacts due to the estimation procedure, yield networks with identical inter-gene edge sets. Here, we discuss further the relationship between DEs and conventional DBNs and present new empirical results on unequally spaced data which demonstrate that modelling Euler approximations in a DBN can lead to improved network reconstruction.

stat.AP↗