Abstract. Current reporting of results based on Markov chain Monte Carlo computations could be improved. In particular, a measure of the accuracy of the resulting estimates is rarely reported. Thus we have little ability to objectively assess the quality of the reported estimates. We address this issue in that we discuss why Monte Carlo standard errors are important, how they can be easily calculated in Markov chain Monte Carlo and how they can be used to decide when to stop the simulation. We compare their use to a popular alternative in the context of two examples.

The viewpoint taken in this paper is that data assimilation is fundamentally a statistical problem and that this problem should be cast in a Bayesian framework. In the absence of model error, the correct solution to the data assimilation problem is to find the posterior distribution implied by this Bayesian setting. Methods for dealing with data assimilation should then be judged by their ability to probe this distribution. In this paper we propose a range of techniques for probing the posterior distribution, based around the Langevin equation; and we compare these new techniques with existing methods. When the underlying dynamics is deterministic, the posterior distribution is on the space of initial conditions leading to a sampling problem over this space. When the underlying dynamics is stochastic the posterior distribution is on the space of continuous time paths. By writing down a density, and conditioning on observations, it is possible to define a range of Markov Chain Monte Carlo (MCMC) methods which sample from the desired posterior distribution, and thereby solve the data assimilation problem. The basic building-blocks for the MCMC methods that we concentrate on in this paper are Langevin equations which are ergodic and whose invariant measures give the desired distribution; in the case of path space sampling these are stochastic partial differential equations (SPDEs). Two examples are given to show how data assimilation can be formulated in a Bayesian fashion. The first is weather prediction, and the second is Lagrangian data assimilation for oceanic velocity fields. Furthermore the relationship between the Bayesian approach outlined here and the commonly used Kalman filter-based techniques, prevalent in practice, is discussed. Two simple pedagogical examples are studied to illustrate the application of Bayesian sampling to data assimilation concretely. Finally a range of open mathematical and computational issues, arising from the Bayesian approach, are outlined.

models with applications to subsurface characterization. The purpose of preconditioning is to reduce the fine-scale computational cost and increase the acceptance rate in the MCMC sampling. This goal is achieved by generating Markov chains based on two-stage computations. In the first stage, a new proposal is first tested by the coarse-scale model based on multiscale finite-volume method. The full fine-scale computation will be conducted only if the proposal passes the coarse-scale screening. For more efficient simulations, an approximation of the full fine-scale computation using pre-computed multiscale basis functions can also be used. Comparing with the regular MCMC method, the preconditioned MCMC method generates a modified Markov chain by incorporating the coarse-scale information of the problem. The conditions under which the modified Markov chain will converge to the correct posterior distribution are stated in the paper. The validity of these assumptions for our application, and the conditions which would guarantee a high acceptance rate are also discussed. We would like to note that coarse-scale models used in the simulations need to be inexpensive, but not necessarily very accurate, as our analysis and numerical simulations demonstrate. We present numerical examples for sampling permeability fields using two-point geostatistics. The Karhunen-Loeve expansion is used to represent the realizations of the permeability field conditioned to the dynamic data, such as production data, as well as some static data. Our numerical examples show that the acceptance rate can be increased by more than ten times if MCMC simulations are preconditioned using coarse-scale models. 1. Introduction. Uncertainties

Abstract. The purpose of the present article is to compare different phase-space sampling methods, such as purely stochastic methods (Rejection method, Metropolized independence sampler, Importance Sampling), stochastically perturbed Molecular Dynamics (Hybrid Monte Carlo, Langevin Dynamics, Biased Random Walk), and purely deterministic methods (Nosé-Hoover chains, Nosé-Poincaré and Recursive Multiple Thermostats (RMT) methods). After recalling some theoretical convergence properties for the various methods, we provide some new convergence results for the Hybrid Monte Carlo scheme, requiring weaker (and easier to check) conditions than previously known conditions. We then turn to the numerical efficiency of the sampling schemes for a benchmark model of linear alkane molecules. In particular, the numerical distributions that are generated are compared in a systematic way, on the basis of some quantitative convergence indicators. 1991 Mathematics Subject Classification. 82B80, 37M25, 65C05, 65C40.

Sequential Monte Carlo (SMC) methods are a class of importance sampling and resampling techniques designed to simulate from a sequence of probability distributions. These approaches have become very popular over the last few years to solve sequential Bayesian inference problems (e.g. Doucet et al. 2001). However, in comparison to Markov chain Monte Carlo (MCMC), the application of SMC remains limited when, in fact, such methods are also appropriate in such contexts (e.g. Chopin (2002); Del Moral et al. (2006)). In this paper, we present a simple unifying framework which allows us to extend both the SMC methodology and its range of applications. Additionally, reinterpreting SMC algorithms as an approximation of nonlinear MCMC kernels, we present alternative SMC and iterative self-interacting approximation (Del Moral & Miclo 2004; 2006) schemes. We demonstrate the performance of the SMC methodology on static and sequential Bayesian inference problems.

In this paper, we present an algorithm for parsing natural images into middle level vision representations – regions, curves, and curve groups (parallel curves and trees). This al-gorithm is targeted for an integrated solution to image segmentation and curve grouping through Bayesian inference. The paper makes the following contributions. (1) It adopts a layered (or 2.1D-sketch) representation integrating both region and curve models which compete to explain an input image. The curve layer occludes the region layer and curves observe a partial order occlusion relation. (2) A Markov chain search scheme Metropolized Gibbs Samplers (MGS) is studied. It consists of several pairs of reversible jumps to tra-verse the complex solution space. An MGS proposes the next state within the jump scope of the current state according to a conditional probability like a Gibbs sampler and then accepts the proposal with a Metropolis-Hastings step. This paper discusses systematic de-sign strategies of devising reversible jumps for a complex inference task. (3) The proposal probability ratios in jumps are factorized into ratios of discriminative probabilities. The latter are computed in a bottom-up process, and they drive the Markov chain dynamics in a data-driven Markov chain Monte Carlo framework. We demonstrate the performance of the algorithm in experiments with a number of natural images.

Kernel models for classification and regression have emerged as widely applied tools in statistics and machine learning. We discuss a Bayesian framework and theory for kernel methods, providing a new rationalisation of kernel regression based on non-parametric Bayesian models. Functional analytic results ensure that such a non-parametric prior specification induces a class of functions that span the reproducing kernel Hilbert space corresponding to the selected kernel. Bayesian analysis of the model allows for direct and formal inference on the uncertain re-gression or classification functions. Augmenting the model with Bayesian vari-able selection priors over kernel bandwidth parameters extends the framework to automatically address the key practical questions of kernel feature selection. Novel, customised MCMC methods are detailed and used in example analyses. The practical benefits and modelling flexibility of the Bayesian kernel framework are illustrated in both simulated and real data examples that address prediction and classification inference with high-dimensional data.

The main goal of this paper is to design an efficient sampling technique for dynamic data integration using the Langevin algorithms. Based on a coarse-scale model of the problem, we compute the proposals of the Langevin algorithms using the coarse-scale gradient of the target distribution. To guarantee a correct and efficient sampling, each proposal is first tested by a Metropolis acceptancerejection step with a coarse-scale distribution. If the proposal is accepted in the first stage, then a fine-scale simulation is performed at the second stage to determine the acceptance probability. Comparing with the direct Langevin algorithm, the new method generates a modified Markov chain by incorporating the coarse-scale information of the problem. Under some mild technical conditions we prove that the modified Markov chain converges to the correct posterior distribution. We would like to note that the coarse-scale models used in the simulations need to be inexpensive, but not necessarily very accurate, as our analysis and numerical simulations demonstrate. We present numerical examples for sampling permeability fields using two-point geostatistics. Karhunen-Loève expansion is used to represent the realizations of the permeability field conditioned to the dynamic data, such as the production data, as well as the static data. The numerical examples show that the coarse-gradient Langevin algorithms are much faster than the direct Langevin algorithms but have similar acceptance rates. 1. Introduction. Uncertainties

Abstract: The basic nonlinear ltering problem for dynamical systems is considered. Approximating the optimal lter estimate by particle lter methods has become perhaps the most common and useful method in recent years. Many variants of particle lters have been suggested, and there is an extensive literature on the theoretical aspects of the quality of the approximation. Still, a clear cut result that the approximate solution, for unbounded functions, converges to the true optimal estimate as the number of particles tends to in nity seems to be lacking. It is the purpose of this contribution to give such a basic convergence result.

The combination of a Markov chain Monte Carlo (MCMC) method with likelihood-based assessment of phylogenies is becoming a popular alternative to direct likelihood optimization. However, MCMC, like maximum likelihood, is a computationallyexpensive method. To approximate the posterior distribution of phylogenies, a Markov chain is constructed, using the Metropolis algorithm, such that the chain has the posterior distribution of the parameters of phylogenies as its stationary distribution. This paper describes parallel algorithms and their MPI-based parallel implementation for MCMC-based Bayesian phylogenetic inference. Bayesian phylogenetic inference is computationally expensive both in time and in memory requirements. Our variations on MCMC and their implementation were done to permit the studyof large phylogenetic problems. In our approach, we can distribute either entire chains or parts of a chain to different processors, since in current models the columns of the data are independent. Evaluations on a 32-node Beowulf cluster suggest the problem scales well. A number of important points are identified, including a superlinear speedup due to more effective cache usage and the point at which additional processors slow down the process due to communication overhead.