arXiv subjects

Matthew M. Dunlop

Publications and source records attributed to Matthew M. Dunlop.

12 recordsLinked to original sources

arXiv2022

A gradient-free subspace-adjusting ensemble sampler for infinite-dimensional Bayesian inverse problems

Sampling of sharp posteriors in high dimensions is a challenging problem, especially when gradients of the likelihood are unavailable. In low to moderate dimensions, affine-invariant methods, a class of ensemble-based gradient-free methods, have found success in sampling concentrated posteriors. However, the number of ensemble members must exceed the dimension of the unknown state in order for the correct distribution to be targeted. Conversely, the preconditioned Crank-Nicolson (pCN) algorithm succeeds at sampling in high dimensions, but samples become highly correlated when the posterior differs significantly from the prior. In this article we combine the above methods in two different ways as an attempt to find a compromise. The first method involves inflating the proposal covariance in pCN with that of the current ensemble, whilst the second performs approximately affine-invariant steps on a continually adapting low-dimensional subspace, while using pCN on its orthogonal complement.

stat.ME↗

arXiv2021

Stability of Gibbs Posteriors from the Wasserstein Loss for Bayesian Full Waveform Inversion

Recently, the Wasserstein loss function has been proven to be effective when applied to deterministic full-waveform inversion (FWI) problems. We consider the application of this loss function in Bayesian FWI so that the uncertainty can be captured in the solution. Other loss functions that are commonly used in practice are also considered for comparison. Existence and stability of the resulting Gibbs posteriors are shown on function space under weak assumptions on the prior and model. In particular, the distribution arising from the Wasserstein loss is shown to be quite stable with respect to high-frequency noise in the data. We then illustrate the difference between the resulting distributions numerically, using Laplace approximations to estimate the unknown velocity field and uncertainty associated with the estimates.

math.ST↗

arXiv2020

Reconciling Bayesian and perimeter regularization for binary inversion

A central theme in classical algorithms for the reconstruction of discontinuous functions from observational data is perimeter regularization via the use of the total variation. On the other hand, sparse or noisy data often demands a probabilistic approach to the reconstruction of images, to enable uncertainty quantification; the Bayesian approach to inversion, which itself introduces a form of regularization, is a natural framework in which to carry this out. In this paper the link between Bayesian inversion methods and perimeter regularization is explored. In this paper two links are studied: (i) the maximum a posteriori (MAP) objective function of a suitably chosen Bayesian phase-field approach is shown to be closely related to a least squares plus perimeter regularization objective; (ii) sample paths of a suitably chosen Bayesian level set formulation are shown to possess finite perimeter and to have the ability to learn about the true perimeter.

math.OC↗

arXiv2019

Multiplicative noise in Bayesian inverse problems: Well-posedness and consistency of MAP estimators

Multiplicative noise arises in inverse problems when, for example, uncertainty on measurements is proportional to the size of the measurement itself. The likelihood that arises is hence more complicated than that from additive noise. We consider two multiplicative noise models: purely multiplicative noise, and a mixture of multiplicative noise and additive noise. Adopting a Bayesian approach, we provide conditions for the resulting posterior distributions on Banach space to be continuous with respect to perturbations in the data; the inclusion of additive noise in addition to multiplicative noise acts as a form of regularization, allowing for milder conditions on the forward map. Additionally, we show that MAP estimators exist for both the purely multiplicative and mixed noise models when a Gaussian prior is employed, and for the latter prove that they are consistent in the small noise limit when all noise is Gaussian.

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arXiv2019

Hyperparameter Estimation in Bayesian MAP Estimation: Parameterizations and Consistency

The Bayesian formulation of inverse problems is attractive for three primary reasons: it provides a clear modelling framework; means for uncertainty quantification; and it allows for principled learning of hyperparameters. The posterior distribution may be explored by sampling methods, but for many problems it is computationally infeasible to do so. In this situation maximum a posteriori (MAP) estimators are often sought. Whilst these are relatively cheap to compute, and have an attractive variational formulation, a key drawback is their lack of invariance under change of parameterization. This is a particularly significant issue when hierarchical priors are employed to learn hyperparameters. In this paper we study the effect of the choice of parameterization on MAP estimators when a conditionally Gaussian hierarchical prior distribution is employed. Specifically we consider the centred parameterization, the natural parameterization in which the unknown state is solved for directly, and the noncentred parameterization, which works with a whitened Gaussian as the unknown state variable, and arises when considering dimension-robust MCMC algorithms; MAP estimation is well-defined in the nonparametric setting only for the noncentred parameterization. However, we show that MAP estimates based on the noncentred parameterization are not consistent as estimators of hyperparameters; conversely, we show that limits of finite-dimensional centred MAP estimators are consistent as the dimension tends to infinity. We also consider empirical Bayesian hyperparameter estimation, show consistency of these estimates, and demonstrate that they are more robust with respect to noise than centred MAP estimates. An underpinning concept throughout is that hyperparameters may only be recovered up to measure equivalence, a well-known phenomenon in the context of the Ornstein-Uhlenbeck process.

math.ST↗

arXiv2019

Dimension-Robust MCMC in Bayesian Inverse Problems

The methodology developed in this article is motivated by a wide range of prediction and uncertainty quantification problems that arise in Statistics, Machine Learning and Applied Mathematics, such as non-parametric regression, multi-class classification and inversion of partial differential equations. One popular formulation of such problems is as Bayesian inverse problems, where a prior distribution is used to regularize inference on a high-dimensional latent state, typically a function or a field. It is common that such priors are non-Gaussian, for example piecewise-constant or heavy-tailed, and/or hierarchical, in the sense of involving a further set of low-dimensional parameters, which, for example, control the scale or smoothness of the latent state. In this formulation prediction and uncertainty quantification relies on efficient exploration of the posterior distribution of latent states and parameters. This article introduces a framework for efficient MCMC sampling in Bayesian inverse problems that capitalizes upon two fundamental ideas in MCMC, non-centred parameterisations of hierarchical models and dimension-robust samplers for latent Gaussian processes. Using a range of diverse applications we showcase that the proposed framework is dimension-robust, that is, the efficiency of the MCMC sampling does not deteriorate as the dimension of the latent state gets higher. We showcase the full potential of the machinery we develop in the article in semi-supervised multi-class classification, where our sampling algorithm is used within an active learning framework to guide the selection of input data to manually label in order to achieve high predictive accuracy with a minimal number of labelled data.

stat.ME↗

arXiv2018

Large Data and Zero Noise Limits of Graph-Based Semi-Supervised Learning Algorithms

Scalings in which the graph Laplacian approaches a differential operator in the large graph limit are used to develop understanding of a number of algorithms for semi-supervised learning; in particular the extension, to this graph setting, of the probit algorithm, level set and kriging methods, are studied. Both optimization and Bayesian approaches are considered, based around a regularizing quadratic form found from an affine transformation of the Laplacian, raised to a, possibly fractional, exponent. Conditions on the parameters defining this quadratic form are identified under which well-defined limiting continuum analogues of the optimization and Bayesian semi-supervised learning problems may be found, thereby shedding light on the design of algorithms in the large graph setting. The large graph limits of the optimization formulations are tackled through $Γ-$convergence, using the recently introduced $TL^p$ metric. The small labelling noise limits of the Bayesian formulations are also identified, and contrasted with pre-existing harmonic function approaches to the problem.

stat.ML↗

arXiv2018

How Deep Are Deep Gaussian Processes?

Recent research has shown the potential utility of Deep Gaussian Processes. These deep structures are probability distributions, designed through hierarchical construction, which are conditionally Gaussian. In this paper, the current published body of work is placed in a common framework and, through recursion, several classes of deep Gaussian processes are defined. The resulting samples generated from a deep Gaussian process have a Markovian structure with respect to the depth parameter, and the effective depth of the resulting process is interpreted in terms of the ergodicity, or non-ergodicity, of the resulting Markov chain. For the classes of deep Gaussian processes introduced, we provide results concerning their ergodicity and hence their effective depth. We also demonstrate how these processes may be used for inference; in particular we show how a Metropolis-within-Gibbs construction across the levels of the hierarchy can be used to derive sampling tools which are robust to the level of resolution used to represent the functions on a computer. For illustration, we consider the effect of ergodicity in some simple numerical examples.

math.ST↗

arXiv2017

Iterative Updating of Model Error for Bayesian Inversion

In computational inverse problems, it is common that a detailed and accurate forward model is approximated by a computationally less challenging substitute. The model reduction may be necessary to meet constraints in computing time when optimization algorithms are used to find a single estimate, or to speed up Markov chain Monte Carlo (MCMC) calculations in the Bayesian framework. The use of an approximate model introduces a discrepancy, or modeling error, that may have a detrimental effect on the solution of the ill-posed inverse problem, or it may severely distort the estimate of the posterior distribution. In the Bayesian paradigm, the modeling error can be considered as a random variable, and by using an estimate of the probability distribution of the unknown, one may estimate the probability distribution of the modeling error and incorporate it into the inversion. We introduce an algorithm which iterates this idea to update the distribution of the model error, leading to a sequence of posterior distributions that are demonstrated empirically to capture the underlying truth with increasing accuracy. Since the algorithm is not based on rejections, it requires only limited full model evaluations. We show analytically that, in the linear Gaussian case, the algorithm converges geometrically fast with respect to the number of iterations. For more general models, we introduce particle approximations of the iteratively generated sequence of distributions; we also prove that each element of the sequence converges in the large particle limit. We show numerically that, as in the linear case, rapid convergence occurs with respect to the number of iterations. Additionally, we show through computed examples that point estimates obtained from this iterative algorithm are superior to those obtained by neglecting the model error.

stat.ME↗

arXiv2016

The Bayesian Formulation of EIT: Analysis and Algorithms

We provide a rigorous Bayesian formulation of the EIT problem in an infinite dimensional setting, leading to well-posedness in the Hellinger metric with respect to the data. We focus particularly on the reconstruction of binary fields where the interface between different media is the primary unknown. We consider three different prior models - log-Gaussian, star-shaped and level set. Numerical simulations based on the implementation of MCMC are performed, illustrating the advantages and disadvantages of each type of prior in the reconstruction, in the case where the true conductivity is a binary field, and exhibiting the properties of the resulting posterior distribution.

math.PR↗

arXiv2016

Hierarchical Bayesian Level Set Inversion

The level set approach has proven widely successful in the study of inverse problems for interfaces, since its systematic development in the 1990s. Recently it has been employed in the context of Bayesian inversion, allowing for the quantification of uncertainty within the reconstruction of interfaces. However the Bayesian approach is very sensitive to the length and amplitude scales in the prior probabilistic model. This paper demonstrates how the scale-sensitivity can be circumvented by means of a hierarchical approach, using a single scalar parameter. Together with careful consideration of the development of algorithms which encode probability measure equivalences as the hierarchical parameter is varied, this leads to well-defined Gibbs based MCMC methods found by alternating Metropolis-Hastings updates of the level set function and the hierarchical parameter. These methods demonstrably outperform non-hierarchical Bayesian level set methods.

math.PR↗

arXiv2016

MAP Estimators for Piecewise Continuous Inversion

We study the inverse problem of estimating a field $u$ from data comprising a finite set of nonlinear functionals of $u$, subject to additive noise; we denote this observed data by $y$. Our interest is in the reconstruction of piecewise continuous fields in which the discontinuity set is described by a finite number of geometric parameters. Natural applications include groundwater flow and electrical impedance tomography. We take a Bayesian approach, placing a prior distribution on $u$ and determining the conditional distribution on $u$ given the data $y$. It is then natural to study maximum a posterior (MAP) estimators. Recently (Dashti et al 2013) it has been shown that MAP estimators can be characterised as minimisers of a generalised Onsager-Machlup functional, in the case where the prior measure is a Gaussian random field. We extend this theory to a more general class of prior distributions which allows for piecewise continuous fields. Specifically, the prior field is assumed to be piecewise Gaussian with random interfaces between the different Gaussians defined by a finite number of parameters. We also make connections with recent work on MAP estimators for linear problems and possibly non-Gaussian priors (Helin, Burger 2015) which employs the notion of Fomin derivative. In showing applicability of our theory we focus on the groundwater flow and EIT models, though the theory holds more generally. Numerical experiments are implemented for the groundwater flow model, demonstrating the feasibility of determining MAP estimators for these piecewise continuous models, but also that the geometric formulation can lead to multiple nearby (local) MAP estimators. We relate these MAP estimators to the behaviour of output from MCMC samples of the posterior, obtained using a state-of-the-art function space Metropolis-Hastings method.

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