arXiv subjects

Alexandros Beskos

Publications and source records attributed to Alexandros Beskos.

At least 37 records · Page 2Linked to original sources

arXiv2022

Bayesian Learning of Graph Substructures

Graphical models provide a powerful methodology for learning the conditional independence structure in multivariate data. Inference is often focused on estimating individual edges in the latent graph. Nonetheless, there is increasing interest in inferring more complex structures, such as communities, for multiple reasons, including more effective information retrieval and better interpretability. Stochastic blockmodels offer a powerful tool to detect such structure in a network. We thus propose to exploit advances in random graph theory and embed them within the graphical models framework. A consequence of this approach is the propagation of the uncertainty in graph estimation to large-scale structure learning. We consider Bayesian nonparametric stochastic blockmodels as priors on the graph. We extend such models to consider clique-based blocks and to multiple graph settings introducing a novel prior process based on a Dependent Dirichlet process. Moreover, we devise a tailored computation strategy of Bayes factors for block structure based on the Savage-Dickey ratio to test for presence of larger structure in a graph. We demonstrate our approach in simulations as well as on real data applications in finance and transcriptomics.

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arXiv2022

Unbiased Estimation using a Class of Diffusion Processes

We study the problem of unbiased estimation of expectations with respect to (w.r.t.) $π$ a given, general probability measure on $(\mathbb{R}^d,\mathcal{B}(\mathbb{R}^d))$ that is absolutely continuous with respect to a standard Gaussian measure. We focus on simulation associated to a particular class of diffusion processes, sometimes termed the Schrödinger-Föllmer Sampler, which is a simulation technique that approximates the law of a particular diffusion bridge process $\{X_t\}_{t\in [0,1]}$ on $\mathbb{R}^d$, $d\in \mathbb{N}_0$. This latter process is constructed such that, starting at $X_0=0$, one has $X_1\sim π$. Typically, the drift of the diffusion is intractable and, even if it were not, exact sampling of the associated diffusion is not possible. As a result, \cite{sf_orig,jiao} consider a stochastic Euler-Maruyama scheme that allows the development of biased estimators for expectations w.r.t.~$π$. We show that for this methodology to achieve a mean square error of $\mathcal{O}(ε^2)$, for arbitrary $ε>0$, the associated cost is $\mathcal{O}(ε^{-5})$. We then introduce an alternative approach that provides unbiased estimates of expectations w.r.t.~$π$, that is, it does not suffer from the time discretization bias or the bias related with the approximation of the drift function. We prove that to achieve a mean square error of $\mathcal{O}(ε^2)$, the associated cost is, with high probability, $\mathcal{O}(ε^{-2}|\log(ε)|^{2+δ})$, for any $δ>0$. We implement our method on several examples including Bayesian inverse problems.

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arXiv2022

A Lagged Particle Filter for Stable Filtering of certain High-Dimensional State-Space Models

We consider the problem of high-dimensional filtering of state-space models (SSMs) at discrete times. This problem is particularly challenging as analytical solutions are typically not available and many numerical approximation methods can have a cost that scales exponentially with the dimension of the hidden state. Inspired by lag-approximation methods for the smoothing problem, we introduce a lagged approximation of the smoothing distribution that is necessarily biased. For certain classes of SSMs, particularly those that forget the initial condition exponentially fast in time, the bias of our approximation is shown to be uniformly controlled in the dimension and exponentially small in time. We develop a sequential Monte Carlo (SMC) method to recursively estimate expectations with respect to our biased filtering distributions. Moreover, we prove for a class of class of SSMs that can contain dependencies amongst coordinates that as the dimension $d\rightarrow\infty$ the cost to achieve a stable mean square error in estimation, for classes of expectations, is of $\mathcal{O}(Nd^2)$ per-unit time, where $N$ is the number of simulated samples in the SMC algorithm. Our methodology is implemented on several challenging high-dimensional examples including the conservative shallow-water model.

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arXiv2022

Manifold Markov chain Monte Carlo methods for Bayesian inference in diffusion models

Bayesian inference for nonlinear diffusions, observed at discrete times, is a challenging task that has prompted the development of a number of algorithms, mainly within the computational statistics community. We propose a new direction, and accompanying methodology, borrowing ideas from statistical physics and computational chemistry, for inferring the posterior distribution of latent diffusion paths and model parameters, given observations of the process. Joint configurations of the underlying process noise and of parameters, mapping onto diffusion paths consistent with observations, form an implicitly defined manifold. Then, by making use of a constrained Hamiltonian Monte Carlo algorithm on the embedded manifold, we are able to perform computationally efficient inference for a class of discretely observed diffusion models. Critically, in contrast with other approaches proposed in the literature, our methodology is highly automated, requiring minimal user intervention and applying alike in a range of settings, including: elliptic or hypo-elliptic systems; observations with or without noise; linear or non-linear observation operators. Exploiting Markovianity, we propose a variant of the method with complexity that scales linearly in the resolution of path discretisation and the number of observation times. Python code reproducing the results is available at https://doi.org/10.5281/zenodo.5796148

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arXiv2021

MCMC Algorithms for Posteriors on Matrix Spaces

We study Markov chain Monte Carlo (MCMC) algorithms for target distributions defined on matrix spaces. Such an important sampling problem has yet to be analytically explored. We carry out a major step in covering this gap by developing the proper theoretical framework that allows for the identification of ergodicity properties of typical MCMC algorithms, relevant in such a context. Beyond the standard Random-Walk Metropolis (RWM) and preconditioned Crank--Nicolson (pCN), a contribution of this paper in the development of a novel algorithm, termed the `Mixed' pCN (MpCN). RWM and pCN are shown not to be geometrically ergodic for an important class of matrix distributions with heavy tails. In contrast, MpCN is robust across targets with different tail behaviour and has very good empirical performance within the class of heavy-tailed distributions. Geometric ergodicity for MpCN is not fully proven in this work, as some remaining drift conditions are quite challenging to obtain owing to the complexity of the state space. We do, however, make a lot of progress towards a proof, and show in detail the last steps left for future work. We illustrate the computational performance of the various algorithms through numerical applications, including calibration on real data of a challenging model arising in financial statistics.

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arXiv2021

Online Smoothing for Diffusion Processes Observed with Noise

We introduce a methodology for online estimation of smoothing expectations for a class of additive functionals, in the context of a rich family of diffusion processes (that may include jumps) -- observed at discrete-time instances. We overcome the unavailability of the transition density of the underlying SDE by working on the augmented pathspace. The new method can be applied, for instance, to carry out online parameter inference for the designated class of models. Algorithms defined on the infinite-dimensional pathspace have been developed in the last years mainly in the context of MCMC techniques. There, the main benefit is the achievement of mesh-free mixing times for the practical time-discretised algorithm used on a PC. Our own methodology sets up the framework for infinite-dimensional online filtering -- an important positive practical consequence is the construct of estimates with the variance that does not increase with decreasing mesh-size. Besides regularity conditions, our method is, in principle, applicable under the weak assumption -- relatively to restrictive conditions often required in the MCMC or filtering literature of methods defined on pathspace -- that the SDE covariance matrix is invertible.

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arXiv2021

Score-Based Parameter Estimation for a Class of Continuous-Time State Space Models

We consider the problem of parameter estimation for a class of continuous-time state space models. In particular, we explore the case of a partially observed diffusion, with data also arriving according to a diffusion process. Based upon a standard identity of the score function, we consider two particle filter based methodologies to estimate the score function. Both methods rely on an online estimation algorithm for the score function of $\mathcal{O}(N^2)$ cost, with $N\in\mathbb{N}$ the number of particles. The first approach employs a simple Euler discretization and standard particle smoothers and is of cost $\mathcal{O}(N^2 + NΔ_l^{-1})$ per unit time, where $Δ_l=2^{-l}$, $l\in\mathbb{N}_0$, is the time-discretization step. The second approach is new and based upon a novel diffusion bridge construction. It yields a new backward type Feynman-Kac formula in continuous-time for the score function and is presented along with a particle method for its approximation. Considering a time-discretization, the cost is $\mathcal{O}(N^2Δ_l^{-1})$ per unit time. To improve computational costs, we then consider multilevel methodologies for the score function. We illustrate our parameter estimation method via stochastic gradient approaches in several numerical examples.

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arXiv2021

A 4D-Var Method with Flow-Dependent Background Covariances for the Shallow-Water Equations

The 4D-Var method for filtering partially observed nonlinear chaotic dynamical systems consists of finding the maximum a-posteriori (MAP) estimator of the initial condition of the system given observations over a time window, and propagating it forward to the current time via the model dynamics. This method forms the basis of most currently operational weather forecasting systems. In practice the optimization becomes infeasible if the time window is too long due to the non-convexity of the cost function, the effect of model errors, and the limited precision of the ODE solvers. Hence the window has to be kept sufficiently short, and the observations in the previous windows can be taken into account via a Gaussian background (prior) distribution. The choice of the background covariance matrix is an important question that has received much attention in the literature. In this paper, we define the background covariances in a principled manner, based on observations in the previous $b$ assimilation windows, for a parameter $b\ge 1$. The method is at most $b$ times more computationally expensive than using fixed background covariances, requires little tuning, and greatly improves the accuracy of 4D-Var. As a concrete example, we focus on the shallow-water equations. The proposed method is compared against state-of-the-art approaches in data assimilation and is shown to perform favourably on simulated data. We also illustrate our approach on data from the recent tsunami of 2011 in Fukushima, Japan.

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arXiv2020

Asymptotic Analysis of Model Selection Criteria for General Hidden Markov Models

The paper obtains analytical results for the asymptotic properties of Model Selection Criteria -- widely used in practice -- for a general family of hidden Markov models (HMMs), thereby substantially extending the related theory beyond typical i.i.d.-like model structures and filling in an important gap in the relevant literature. In particular, we look at the Bayesian and Akaike Information Criteria (BIC and AIC) and the model evidence. In the setting of nested classes of models, we prove that BIC and the evidence are strongly consistent for HMMs (under regularity conditions), whereas AIC is not weakly consistent. Numerical experiments support our theoretical results.

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arXiv2019

Monte Carlo Co-Ordinate Ascent Variational Inference

In Variational Inference (VI), coordinate-ascent and gradient-based approaches are two major types of algorithms for approximating difficult-to-compute probability densities. In real-world implementations of complex models, Monte Carlo methods are widely used to estimate expectations in coordinate-ascent approaches and gradients in derivative-driven ones. We discuss a Monte Carlo Co-ordinate Ascent VI (MC-CAVI) algorithm that makes use of Markov chain Monte Carlo (MCMC) methods in the calculation of expectations required within Co-ordinate Ascent VI (CAVI). We show that, under regularity conditions, an MC-CAVI recursion will get arbitrarily close to a maximiser of the evidence lower bound (ELBO) with any given high probability. In numerical examples, the performance of MC-CAVI algorithm is compared with that of MCMC and -- as a representative of derivative-based VI methods -- of Black Box VI (BBVI). We discuss and demonstrate MC-CAVI's suitability for models with hard constraints in simulated and real examples. We compare MC-CAVI's performance with that of MCMC in an important complex model used in Nuclear Magnetic Resonance (NMR) spectroscopy data analysis -- BBVI is nearly impossible to be employed in this setting due to the hard constraints involved in the model.

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arXiv2018

Particle Filtering for Stochastic Navier-Stokes Signal Observed with Linear Additive Noise

We consider a non-linear filtering problem, whereby the signal obeys the stochastic Navier-Stokes equations and is observed through a linear mapping with additive noise. The setup is relevant to data assimilation for numerical weather prediction and climate modelling, where similar models are used for unknown ocean or wind velocities. We present a particle filtering methodology that uses likelihood informed importance proposals, adaptive tempering, and a small number of appropriate Markov Chain Monte Carlo steps. We provide a detailed design for each of these steps and show in our numerical examples that they are all crucial in terms of achieving good performance and efficiency.

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arXiv2018

Optimization Based Methods for Partially Observed Chaotic Systems

In this paper we consider filtering and smoothing of partially observed chaotic dynamical systems that are discretely observed, with an additive Gaussian noise in the observation. These models are found in a wide variety of real applications and include the Lorenz 96' model. In the context of a fixed observation interval $T$, observation time step $h$ and Gaussian observation variance $σ_Z^2$, we show under assumptions that the filter and smoother are well approximated by a Gaussian with high probability when $h$ and $σ^2_Z h$ are sufficiently small. Based on this result we show that the Maximum-a-posteriori (MAP) estimators are asymptotically optimal in mean square error as $σ^2_Z h$ tends to $0$. Given these results, we provide a batch algorithm for the smoother and filter, based on Newton's method, to obtain the MAP. In particular, we show that if the initial point is close enough to the MAP, then Newton's method converges to it at a fast rate. We also provide a method for computing such an initial point. These results contribute to the theoretical understanding of widely used 4D-Var data assimilation method. Our approach is illustrated numerically on the Lorenz 96' model with state vector up to 1 million dimensions, with code running in the order of minutes. To our knowledge the results in this paper are the first of their type for this class of models.

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arXiv2018

On Concentration Properties of Partially Observed Chaotic Systems

This article presents results on the concentration properties of the smoothing and filtering distributions of some partially observed chaotic dynamical systems. We show that, rather surprisingly, for the geometric model of the Lorenz equations, as well as some other chaotic dynamical systems, the smoothing and filtering distributions do not concentrate around the true position of the signal, as the number of observations tends to infinity. Instead, under various assumptions on the observation noise, we show that the expected value of the diameter of the support of the smoothing and filtering distributions remains lower bounded by a constant times the standard deviation of the noise, independently of the number of observations. Conversely, under rather general conditions, the diameter of the support of the smoothing and filtering distributions are upper bounded by a constant times the standard deviation of the noise. To some extent, applications to the three dimensional Lorenz 63' model and to the Lorenz 96' model of arbitrarily large dimension are considered.

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arXiv2017

Efficient sequential Monte Carlo algorithms for integrated population models

State-space models are commonly used to describe different forms of ecological data. We consider the case of count data with observation errors. For such data the system process is typically multi-dimensional consisting of coupled Markov processes, where each component corresponds to a different characterisation of the population, such as age group, gender or breeding status. The associated system process equations describe the biological mechanisms under which the system evolves over time. However, there is often limited information in the count data alone to sensibly estimate demographic parameters of interest, so these are often combined with additional ecological observations leading to an integrated data analysis. Unfortunately, fitting these models to the data can be challenging, especially if the state-space model for the count data is non-linear or non-Gaussian. We propose an efficient particle Markov chain Monte Carlo algorithm to estimate the demographic parameters without the need for resorting to linear or Gaussian approximations. In particular, we exploit the integrated model structure to enhance the efficiency of the algorithm. We then incorporate the algorithm into a sequential Monte Carlo sampler in order to perform model comparison with regards to the dependence structure of the demographic parameters. Finally, we demonstrate the applicability and computational efficiency of our algorithms on two real datasets.

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arXiv2017

Multilevel Sequential Monte Carlo with Dimension-Independent Likelihood-Informed Proposals

In this article we develop a new sequential Monte Carlo (SMC) method for multilevel (ML) Monte Carlo estimation. In particular, the method can be used to estimate expectations with respect to a target probability distribution over an infinite-dimensional and non-compact space as given, for example, by a Bayesian inverse problem with Gaussian random field prior. Under suitable assumptions the MLSMC method has the optimal $O(ε^{-2})$ bound on the cost to obtain a mean-square error of $O(ε^2)$. The algorithm is accelerated by dimension-independent likelihood-informed (DILI) proposals designed for Gaussian priors, leveraging a novel variation which uses empirical sample covariance information in lieu of Hessian information, hence eliminating the requirement for gradient evaluations. The efficiency of the algorithm is illustrated on two examples: inversion of noisy pressure measurements in a PDE model of Darcy flow to recover the posterior distribution of the permeability field, and inversion of noisy measurements of the solution of an SDE to recover the posterior path measure.

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arXiv2017

Geometric MCMC for Infinite-Dimensional Inverse Problems

Bayesian inverse problems often involve sampling posterior distributions on infinite-dimensional function spaces. Traditional Markov chain Monte Carlo (MCMC) algorithms are characterized by deteriorating mixing times upon mesh-refinement, when the finite-dimensional approximations become more accurate. Such methods are typically forced to reduce step-sizes as the discretization gets finer, and thus are expensive as a function of dimension. Recently, a new class of MCMC methods with mesh-independent convergence times has emerged. However, few of them take into account the geometry of the posterior informed by the data. At the same time, recently developed geometric MCMC algorithms have been found to be powerful in exploring complicated distributions that deviate significantly from elliptic Gaussian laws, but are in general computationally intractable for models defined in infinite dimensions. In this work, we combine geometric methods on a finite-dimensional subspace with mesh-independent infinite-dimensional approaches. Our objective is to speed up MCMC mixing times, without significantly increasing the computational cost per step (for instance, in comparison with the vanilla preconditioned Crank-Nicolson (pCN) method). This is achieved by using ideas from geometric MCMC to probe the complex structure of an intrinsic finite-dimensional subspace where most data information concentrates, while retaining robust mixing times as the dimension grows by using pCN-like methods in the complementary subspace. The resulting algorithms are demonstrated in the context of three challenging inverse problems arising in subsurface flow, heat conduction and incompressible flow control. The algorithms exhibit up to two orders of magnitude improvement in sampling efficiency when compared with the pCN method.

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arXiv2015

Asymptotic Analysis of the Random-Walk Metropolis Algorithm on Ridged Densities

In this paper we study the asymptotic behavior of the Random-Walk Metropolis algorithm on probability densities with two different `scales', where most of the probability mass is distributed along certain key directions with the `orthogonal' directions containing relatively less mass. Such class of probability measures arise in various applied contexts including Bayesian inverse problems where the posterior measure concentrates on a sub-manifold when the noise variance goes to zero. When the target measure concentrates on a linear sub-manifold, we derive analytically a diffusion limit for the Random-Walk Metropolis Markov chain as the scale parameter goes to zero. In contrast to the existing works on scaling limits, our limiting Stochastic Differential Equation does not in general have a constant diffusion coefficient. Our results show that in some cases, the usual practice of adapting the step-size to control the acceptance probability might be sub-optimal as the optimal acceptance probability is zero (in the limit).

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arXiv2015

Bayesian Inference for Duplication-Mutation with Complementarity Network Models

We observe an undirected graph $G$ without multiple edges and self-loops, which is to represent a protein-protein interaction (PPI) network. We assume that $G$ evolved under the duplication-mutation with complementarity (DMC) model from a seed graph, $G_0$, and we also observe the binary forest $Γ$ that represents the duplication history of $G$. A posterior density for the DMC model parameters is established, and we outline a sampling strategy by which one can perform Bayesian inference; that sampling strategy employs a particle marginal Metropolis-Hastings (PMMH) algorithm. We test our methodology on numerical examples to demonstrate a high accuracy and precision in the inference of the DMC model's mutation and homodimerization parameters.

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