Alexandros Beskos

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.

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 . by using separable Hamiltonian dynamics with potential -log .. 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 O(1) acceptance probability as the dimension d of the state space tends to ., the leapfrog step-size h should be scaled as h=l ×d−1/ 4 . Therefore, in high dimensions, HMC requires O(d1/ 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 decreases as l increases (because fewer steps are required to reach the desired final integration time), against the cost related to the average number of proposals required to obtain acceptance, which increases as l increases

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 Rd for large d. It is well known [Bengtsson, Bickel and Li, In Probability and Statistics: Essays in Honor of David A. Freedman, D. Nolan and T. Speed, eds. (2008) 316–334 IMS; see also Pushing the Limits of Contemporary Statistics (2008) 318–32 9 IMS, Mon. Weather Rev. (2009) 136 (2009) 4629–4640] 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; see, for example, Chopin [Biometrika 89 (2002) 539–551]; see also [J. R. Stat. Soc. Ser. B Stat. Methodol. 68 (2006) 411–436, Phys. Rev. Lett. 78 (1997) 2690–2693, Stat. Comput. 11 (2001) 125–139]. 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 eN as d → ∞ with 1 < eN < N. The convergence is achieved with a computational cost proportional to Nd2. If eN � N, we can raise its value by introducing a number of resampling steps, say m (where m is independent of d). In this case, the ESS converges to a random variable eN,m as d → ∞ and limm→∞ eN,m = N. Also, we show that the Monte Carlo error for estimating a fixed-dimensional marginal expectation is of order √ 1 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 resampling leads to a reduction in the Monte Carlo error and increase in the ESS. All of our analysis is made under the assumption that the target density is i.i.d.

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

MCMC methods for diffusion bridges

n

MONTE-CARLO MAXIMUM LIKELIHOOD ESTIMATION FOR DISCRETELY OBSERVED DIFFUSION PROCESSES

of the state space, asymptotically as

MCMC methods for sampling function space

n\to\infty

Error bounds and normalising constants for sequential Monte Carlo samplers in high dimensions

. 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.

Sequential Monte Carlo Methods for High-Dimensional Inverse Problems: A Case Study for the Navier-Stokes Equations ∗

We present and study a Langevin MCMC approach for sampling nonlinear diffusion bridges. The method is based on recent theory concerning stochastic partial differential equations (SPDEs) reversible with respect to the target bridge, derived by applying the Langevin idea on the bridge pathspace. In the process, a Random-Walk Metropolis algorithm and an Independence Sampler are also obtained. The novel algorithmic idea of the paper is that proposed moves for the MCMC algorithm are determined by discretising the SPDEs in the time direction using an implicit scheme, parametrised by θ ∈ [0,1]. We show that the resulting infinite-dimensional MCMC sampler is well-defined only if θ = 1/2, when the MCMC proposals have the correct quadratic variation. Previous Langevin-based MCMC methods used explicit schemes, corresponding to θ = 0. The significance of the choice θ = 1/2 is inherited by the finite-dimensional approximation of the algorithm used in practice. We present numerical results illustrating the phenomenon and the theory that explains it. Diffusion bridges (with additive noise) are representative of the family of laws defined as a change of measure from Gaussian distributions on arbitrary separable Hilbert spaces; the analysis in this paper can be readily extended to target laws from this family and an example from signal processing illustrates this fact.

Multilevel sequential Monte Carlo samplers

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 aid 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 -> infinity, we show that the number of Monte Carlo iterations should be tuned as O (n(1/2)) and we demonstrate the consistency properties of the Monte Carlo MLE as an estimator of the true parameter value.