arXiv subjects

Alexandros Beskos

Publications and source records attributed to Alexandros Beskos.

50 records · Page 3Linked to original sources

arXiv2015

Multilevel Sequential Monte Carlo Samplers

In this article we consider the approximation of expectations w.r.t. probability distributions associated to the solution of partial differential equations (PDEs); this scenario appears routinely in Bayesian inverse problems. In practice, one often has to solve the associated PDE numerically, using, for instance finite element methods and leading to a discretisation bias, with the step-size level $h_L$. In addition, the expectation cannot be computed analytically and one often resorts to Monte Carlo methods. In the context of this problem, it is known that the introduction of the multilevel Monte Carlo (MLMC) method can reduce the amount of computational effort to estimate expectations, for a given level of error. This is achieved via a telescoping identity associated to a Monte Carlo approximation of a sequence of probability distributions with discretisation levels $\infty>h_0>h_1\cdots>h_L$. In many practical problems of interest, one cannot achieve an i.i.d. sampling of the associated sequence of probability distributions. A sequential Monte Carlo (SMC) version of the MLMC method is introduced to deal with this problem. It is shown that under appropriate assumptions, the attractive property of a reduction of the amount of computational effort to estimate expectations, for a given level of error, can be maintained within the SMC context.

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arXiv2015

Bayesian Inference for partially observed SDEs Driven by Fractional Brownian Motion

We consider continuous-time diffusion models driven by fractional Brownian motion. Observations are assumed to possess a non-trivial likelihood given the latent path. Due to the non-Markovianity and high-dimensionality of the latent paths, estimating posterior expectations is a computationally challenging undertaking. We present a reparameterization framework based on the Davies and Harte method for sampling stationary Gaussian processes and use this framework to construct a Markov chain Monte Carlo algorithm that allows computationally efficient Bayesian inference. The Markov chain Monte Carlo algorithm is based on a version of hybrid Monte Carlo that delivers increased efficiency when applied on the high-dimensional latent variables arising in this context. We specify the methodology on a stochastic volatility model allowing for memory in the volatility increments through a fractional specification. The methodology is illustrated on simulated data and on the S&P500/VIX time series and is shown to be effective. Contrary to a long range dependence attribute of such models often assumed in the literature, with Hurst parameter larger than 1/2, the posterior distribution favours values smaller than 1/2, pointing towards medium range dependence.

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arXiv2014

A simulation approach for change-points on phylogenetic trees

We observe $n$ sequences at each of $m$ sites, and assume that they have evolved from an ancestral sequence that forms the root of a binary tree of known topology and branch lengths, but the sequence states at internal nodes are unknown. The topology of the tree and branch lengths are the same for all sites, but the parameters of the evolutionary model can vary over sites. We assume a piecewise constant model for these parameters, with an unknown number of change-points and hence a trans-dimensional parameter space over which we seek to perform Bayesian inference. We propose two novel ideas to deal with the computational challenges of such inference. Firstly, we approximate the model based on the time machine principle: the top nodes of the binary tree (near the root) are replaced by an approximation of the true distribution; as more nodes are removed from the top of the tree, the cost of computing the likelihood is reduced linearly in $n$. The approach introduces a bias, which we investigate empirically. Secondly, we develop a particle marginal Metropolis-Hastings (PMMH) algorithm, that employs a sequential Monte Carlo (SMC) sampler and can use the first idea. Our time-machine PMMH algorithm copes well with one of the bottle-necks of standard computational algorithms: the trans-dimensional nature of the posterior distribution. The algorithm is implemented on simulated and real data examples, and we empirically demonstrate its potential to outperform competing methods based on approximate Bayesian computation (ABC) techniques.

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arXiv2014

A Stable Manifold MCMC Method for High Dimensions

We combine two important recent advancements of MCMC algorithms: first, methods utilizing the intrinsic manifold structure of the parameter space; then, algorithms effective for targets in infinite-dimensions with the critical property that their mixing time is robust to mesh refinement.

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arXiv2014

On the Convergence of Adaptive Sequential Monte Carlo Methods

In several implementations of Sequential Monte Carlo (SMC) methods it is natural, and important in terms of algorithmic efficiency, to exploit the information of the history of the samples to optimally tune their subsequent propagations. In this article we provide a carefully formulated asymptotic theory for a class of such \emph{adaptive} SMC methods. The theoretical framework developed here will cover, under assumptions, several commonly used SMC algorithms. There are only limited results about the theoretical underpinning of such adaptive methods: we will bridge this gap by providing a weak law of large numbers (WLLN) and a central limit theorem (CLT) for some of these algorithms. The latter seems to be the first result of its kind in the literature and provides a formal justification of algorithms used in many real data context. We establish that for a general class of adaptive SMC algorithms the asymptotic variance of the estimators from the adaptive SMC method is \emph{identical} to a so-called `perfect' SMC algorithm which uses ideal proposal kernels. Our results are supported by application on a complex high-dimensional posterior distribution associated with the Navier-Stokes model, where adapting high-dimensional parameters of the proposal kernels is critical for the efficiency of the algorithm.

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arXiv2013

Sequential Monte Carlo Methods for High-Dimensional Inverse Problems: A case study for the Navier-Stokes equations

We consider the inverse problem of estimating the initial condition of a partial differential equation, which is only observed through noisy measurements at discrete time intervals. In particular, we focus on the case where Eulerian measurements are obtained from the time and space evolving vector field, whose evolution obeys the two-dimensional Navier-Stokes equations defined on a torus. This context is particularly relevant to the area of numerical weather forecasting and data assimilation. We will adopt a Bayesian formulation resulting from a particular regularization that ensures the problem is well posed. In the context of Monte Carlo based inference, it is a challenging task to obtain samples from the resulting high dimensional posterior on the initial condition. In real data assimilation applications it is common for computational methods to invoke the use of heuristics and Gaussian approximations. The resulting inferences are biased and not well-justified in the presence of non-linear dynamics and observations. On the other hand, Monte Carlo methods can be used to assimilate data in a principled manner, but are often perceived as inefficient in this context due to the high-dimensionality of the problem. In this work we will propose a generic Sequential Monte Carlo (SMC) sampling approach for high dimensional inverse problems that overcomes these difficulties. The method builds upon Markov chain Monte Carlo (MCMC) techniques, which are currently considered as benchmarks for evaluating data assimilation algorithms used in practice. In our numerical examples, the proposed SMC approach achieves the same accuracy as MCMC but in a much more efficient manner.

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arXiv2013

Advanced MCMC Methods for Sampling on Diffusion Pathspace

The need to calibrate increasingly complex statistical models requires a persistent effort for further advances on available, computationally intensive Monte Carlo methods. We study here an advanced version of familiar Markov Chain Monte Carlo (MCMC) algorithms that sample from target distributions defined as change of measures from Gaussian laws on general Hilbert spaces. Such a model structure arises in several contexts: we focus here at the important class of statistical models driven by diffusion paths whence the Wiener process constitutes the reference Gaussian law. Particular emphasis is given on advanced Hybrid Monte-Carlo (HMC) which makes large, derivative-driven steps in the state space (in contrast with local-move Random-walk-type algorithms) with analytical and experimental results. We illustrate it's computational advantages in various diffusion processes and observation regimes; examples include stochastic volatility and latent survival models. In contrast with their standard MCMC counterparts, the advanced versions have mesh-free mixing times, as these will not deteriorate upon refinement of the approximation of the inherently infinite-dimensional diffusion paths by finite-dimensional ones used in practice when applying the algorithms on a computer.

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arXiv2012

ε-Strong simulation of the Brownian path

We present an iterative sampling method which delivers upper and lower bounding processes for the Brownian path. We develop such processes with particular emphasis on being able to unbiasedly simulate them on a personal computer. The dominating processes converge almost surely in the supremum and $L_1$ norms. In particular, the rate of converge in $L_1$ is of the order $\mathcal {O}(\mathcal{K}^{-1/2})$, $\mathcal{K}$ denoting the computing cost. The a.s. enfolding of the Brownian path can be exploited in Monte Carlo applications involving Brownian paths whence our algorithm (termed the $\varepsilon$-strong algorithm) can deliver unbiased Monte Carlo estimators over path expectations, overcoming discretisation errors characterising standard approaches. We will show analytical results from applications of the $\varepsilon$-strong algorithm for estimating expectations arising in option pricing. We will also illustrate that individual steps of the algorithm can be of separate interest, giving new simulation methods for interesting Brownian distributions.

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arXiv2012

On the Stability of Sequential Monte Carlo Methods in High Dimensions

We investigate the stability of a Sequential Monte Carlo (SMC) method applied to the problem of sampling from a target distribution on $\mathbb{R}^d$ for large $d$. It is well known that using a single importance sampling step one produces an approximation for the target that deteriorates as the dimension $d$ increases, unless the number of Monte Carlo samples $N$ increases at an exponential rate in $d$. We show that this degeneracy can be avoided by introducing a sequence of artificial targets, starting from a `simple' density and moving to the one of interest, using an SMC method to sample from the sequence. Using this class of SMC methods with a fixed number of samples, one can produce an approximation for which the effective sample size (ESS) converges to a random variable $\varepsilon_N$ as $d\rightarrow\infty$ with $1<\varepsilon_{N}<N$. The convergence is achieved with a computational cost proportional to $Nd^2$. If $\varepsilon_N\ll N$, we can raise its value by introducing a number of resampling steps, say $m$ (where $m$ is independent of $d$). In this case, ESS converges to a random variable $\varepsilon_{N,m}$ as $d\rightarrow\infty$ and $\lim_{m\to\infty}\varepsilon_{N,m}=N$. Also, we show that the Monte Carlo error for estimating a fixed dimensional marginal expectation is of order $\frac{1}{\sqrt{N}}$ uniformly in $d$. The results imply that, in high dimensions, SMC algorithms can efficiently control the variability of the importance sampling weights and estimate fixed dimensional marginals at a cost which is less than exponential in $d$ and indicate that, in high dimensions, resampling leads to a reduction in the Monte Carlo error and increase in the ESS.

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arXiv2011

Error Bounds and Normalizing Constants for Sequential Monte Carlo in High Dimensions

In a recent paper Beskos et al (2011), the Sequential Monte Carlo (SMC) sampler introduced in Del Moral et al (2006), Neal (2001) has been shown to be asymptotically stable in the dimension of the state space d at a cost that is only polynomial in d, when N the number of Monte Carlo samples, is fixed. More precisely, it has been established that the effective sample size (ESS) of the ensuing (approximate) sample and the Monte Carlo error of fixed dimensional marginals will converge as $d$ grows, with a computational cost of $\mathcal{O}(Nd^2)$. In the present work, further results on SMC methods in high dimensions are provided as $d\to\infty$ and with $N$ fixed. We deduce an explicit bound on the Monte-Carlo error for estimates derived using the SMC sampler and the exact asymptotic relative $\mathbb{L}_2$-error of the estimate of the normalizing constant. We also establish marginal propagation of chaos properties of the algorithm. The accuracy in high-dimensions of some approximate SMC-based filtering schemes is also discussed.

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arXiv2010

Optimal tuning of the Hybrid Monte-Carlo Algorithm

We investigate the properties of the Hybrid Monte-Carlo algorithm (HMC) in high dimensions. HMC develops a Markov chain reversible w.r.t. a given target distribution $Π$ by using separable Hamiltonian dynamics with potential $-\logΠ$. The additional momentum variables are chosen at random from the Boltzmann distribution and the continuous-time Hamiltonian dynamics are then discretised using the leapfrog scheme. The induced bias is removed via a Metropolis-Hastings accept/reject rule. In the simplified scenario of independent, identically distributed components, we prove that, to obtain an $\mathcal{O}(1)$ acceptance probability as the dimension $d$ of the state space tends to $\infty$, the leapfrog step-size $h$ should be scaled as $h= l \times d^{-1/4}$. Therefore, in high dimensions, HMC requires $\mathcal{O}(d^{1/4})$ steps to traverse the state space. We also identify analytically the asymptotically optimal acceptance probability, which turns out to be 0.651 (to three decimal places). This is the choice which optimally balances the cost of generating a proposal, which {\em decreases} as $l$ increases, against the cost related to the average number of proposals required to obtain acceptance, which {\em increases} as $l$ increases.

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arXiv2009

Optimal scalings for local Metropolis--Hastings chains on nonproduct targets in high dimensions

We investigate local MCMC algorithms, namely the random-walk Metropolis and the Langevin algorithms, and identify the optimal choice of the local step-size as a function of the dimension $n$ of the state space, asymptotically as $n\to\infty$. We consider target distributions defined as a change of measure from a product law. Such structures arise, for instance, in inverse problems or Bayesian contexts when a product prior is combined with the likelihood. We state analytical results on the asymptotic behavior of the algorithms under general conditions on the change of measure. Our theory is motivated by applications on conditioned diffusion processes and inverse problems related to the 2D Navier--Stokes equation.

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arXiv2009

Monte Carlo maximum likelihood estimation for discretely observed diffusion processes

This paper introduces a Monte Carlo method for maximum likelihood inference in the context of discretely observed diffusion processes. The method gives unbiased and a.s.\@ continuous estimators of the likelihood function for a family of diffusion models and its performance in numerical examples is computationally efficient. It uses a recently developed technique for the exact simulation of diffusions, and involves no discretization error. We show that, under regularity conditions, the Monte Carlo MLE converges a.s. to the true MLE. For datasize $n\to\infty$, we show that the number of Monte Carlo iterations should be tuned as $\mathcal{O}(n^{1/2})$ and we demonstrate the consistency properties of the Monte Carlo MLE as an estimator of the true parameter value.

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arXiv2006

Exact simulation of diffusions

We describe a new, surprisingly simple algorithm, that simulates exact sample paths of a class of stochastic differential equations. It involves rejection sampling and, when applicable, returns the location of the path at a random collection of time instances. The path can then be completed without further reference to the dynamics of the target process.

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