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Adaptive reduced tempering For Bayesian inverse problems and rare event simulation
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Abstract
This work proposes an adaptive sequential Monte Carlo sampling algorithm for solving inverse Bayesian problems in a context where a (costly) likelihood evaluation can be approximated by a surrogate, constructed from previous evaluations of the true likelihood. A rough error estimation of the obtained surrogates is required. The method is based on an adaptive sequential Monte-Carlo (SMC) simulation that jointly adapts the likelihood approximations and a standard tempering scheme of the target posterior distribution. This algorithm is well-suited to cases where the posterior is concentrated in a rare and unknown region of the prior. It is also suitable for solving low-temperature and rare-event simulation problems. The main contribution is to propose an entropy criteria that associates to the accuracy of the current surrogate a maximum inverse temperature for the likelihood approximation. The latter is used to sample a so-called snapshot, perform an exact likelihood evaluation, and update the surrogate and its error quantification. Some consistency results are presented in an idealized framework of the proposed algorithm. Our numerical experiments use in particular a reduced basis approach to construct approximate parametric solutions of a partially observed solution of an elliptic Partial Differential Equation. They demonstrate the convergence of the algorithm and show a significant cost reduction (close to a factor 10) for comparable accuracy.
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Atmospheric motion vectors (AMVs) extracted from satellite imagery are the only wind observations with good global coverage. They are important features for feeding numerical weather prediction (NWP) models. Several Bayesian models have been proposed to estimate AMVs. Although critical for correct assimilation into NWP models, very few methods provide a thorough characterization of the estimation errors. The difficulty of estimating errors stems from the specificity of the posterior distribution, which is both very high dimensional, and highly ill-conditioned due to a singular likelihood, which becomes critical in particular in the case of missing data (unobserved pixels). Motivated by this difficult inverse problem, this work studies the evaluation of the (expected) estimation errors using gradient-based Markov chain Monte Carlo (MCMC) algorithms. The main contribution is to propose a general strategy, called here ‘chilling’, which amounts to sampling a local approximation of the posterior distribution in the neighborhood of a point estimate. From a theoretical point of view, we show that under regularity assumptions, the family of chilled posterior distributions converges in distribution as temperature decreases to an optimal Gaussian approximation at a point estimate given by the maximum a posteriori, also known as the Laplace approximation. Chilled sampling therefore provides access to this approximation generally out of reach in such high-dimensional nonlinear contexts. From an empirical perspective, we evaluate the proposed approach based on some quantitative Bayesian criteria. Our numerical simulations are performed on synthetic and real meteorological data. They reveal that not only the proposed chilling exhibits a significant gain in terms of accuracy of the AMV point estimates and of their associated expected error estimates, but also a substantial acceleration in the convergence speed of the MCMC algorithms.
Active learning methods have recently surged in the literature due to their ability to solve complex structural reliability problems within an affordable computational cost. These methods are designed by adaptively building an inexpensive surrogate of the original limit-state function. Examples of such surrogates include Gaussian process models which have been adopted in many contributions, the most popular ones being the efficient global reliability analysis (EGRA) and the active Kriging Monte Carlo simulation (AK-MCS), two milestone contributions in the field. In this paper, we first conduct a survey of the recent literature, showing that most of the proposed methods actually span from modifying one or more aspects of the two aforementioned methods. We then propose a generalized modular framework to build on-the-fly efficient active learning strategies by combining the following four ingredients or modules: surrogate model, reliability estimation algorithm, learning function and stopping criterion. Using this framework, we devise 39 strategies for the solution of 20 reliability benchmark problems. The results of this extensive benchmark (more than 12,000 reliability problems solved) are analyzed under various criteria leading to a synthesized set of recommendations for practitioners. These may be refined with a priori knowledge about the feature of the problem to solve, i.e. dimensionality and magnitude of the failure probability. This benchmark has eventually highlighted the importance of using surrogates in conjunction with sophisticated reliability estimation algorithms as a way to enhance the efficiency of the latter.
In the current work we present two generalizations of the Parallel Tempering algorithm in the context of discrete-time Markov chain Monte Carlo methods for Bayesian inverse problems. These generalizations use state-dependent swapping rates, inspired by the so-called continuous time Infinite Swapping algorithm presented in Plattner et al. (J Chem Phys 135(13):134111, 2011). We analyze the reversibility and ergodicity properties of our generalized PT algorithms. Numerical results on sampling from different target distributions, show that the proposed methods significantly improve sampling efficiency over more traditional sampling algorithms such as Random Walk Metropolis, preconditioned Crank–Nicolson, and (standard) Parallel Tempering.
The identification of parameters in mathematical models using noisy observations is a common task in uncertainty quantification. We employ the framework of Bayesian inversion: we combine monitoring and observational data with prior information to estimate the posterior distribution of a parameter. Specifically, we are interested in the distribution of a diffusion coefficient of an elliptic PDE. In this setting, the sample space is high-dimensional, and each sample of the PDE solution is expensive. To address these issues we propose and analyse a novel Sequential Monte Carlo (SMC) sampler for the approximation of the posterior distribution. Classical, single-level SMC constructs a sequence of measures, starting with the prior distribution, and finishing with the posterior distribution. The intermediate measures arise from a tempering of the liklihood, or, equivalently, a rescaling of the noise. The resolution of the PDE discretisation is fixed. In contrast, our estimator employs a hierarchy of PDE discretisations to decrease the computational cost. We construct a sequence of intermediate measures by decreasing the temperature or by increasing the discretisation level at the same time. This idea builds on and generalises the multi-resolution sampler proposed in [P.S. Koutsourelakis, J. Comput. Phys., 228 (2009), pp. 6184-6211] where a bridging scheme is used to transfer samples from coarse to fine discretisation levels. Importantly, our choice between tempering and bridging is fully adaptive. We present numerical experiments in 2D space, comparing our estimator to single-level SMC and the multi-resolution sampler.
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 when the data is finite dimensional. 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 under a simplifying assumption. 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.
Performing Bayesian inference via Markov chain Monte Carlo (MCMC) can be exceedingly expensive when posterior evaluations invoke the evaluation of a computationally expensive model, such as a system of partial differential equations. In recent work [Conrad et al. JASA 2015, arXiv:1402.1694] we described a framework for constructing and refining local approximations of such models during an MCMC simulation. These posterior--adapted approximations harness regularity of the model to reduce the computational cost of inference while preserving asymptotic exactness of the Markov chain. Here we describe two extensions of that work. First, focusing on the Metropolis--adjusted Langevin algorithm, we describe how a proposal distribution can successfully employ gradients and other relevant information extracted from the approximation. Second, we prove that samplers running in parallel can collaboratively construct a shared posterior approximation while ensuring ergodicity of each associated chain, providing a novel opportunity for exploiting parallel computation in MCMC. We investigate the practical performance of our strategies using two challenging inference problems, the first in subsurface hydrology and the second in glaciology. Using local approximations constructed via parallel chains, we successfully reduce the run time needed to characterize the posterior distributions in these problems from days to hours and from months to days, respectively, dramatically improving the tractability of Bayesian inference.
We consider the Riemann manifold Hamiltonian Monte Carlo (RMHMC) method for solving statistical inverse problems governed by partial differential equations (PDEs). The Bayesian framework is employed to cast the inverse problem into the task of statistical inference whose solution is the posterior distribution in infinite dimensional parameter space conditional upon observation data and Gaussian prior measure. We discretize both the likelihood and the prior using the H¹-conforming finite element method together with a matrix transfer technique. The power of the RMHMC method is that it exploits the geometric structure induced by the PDE constraints of the underlying inverse problem. Consequently, each RMHMC posterior sample is almost uncorrelated/independent from the others providing statistically efficient Markov chain simulation. However this statistical efficiency comes at a computational cost. This motivates us to consider computationally more efficient strategies for RMHMC. At the heart of our construction is the fact that for Gaussian error structures the Fisher information matrix coincides with the Gauss–Newton Hessian. We exploit this fact in considering a computationally simplified RMHMC method combining state-of-the-art adjoint techniques and the superiority of the RMHMC method. Specifically, we first form the Gauss–Newton Hessian at the maximum a posteriori point and then use it as a fixed constant metric tensor throughout RMHMC simulation. This eliminates the need for the computationally costly differential geometric Christoffel symbols, which in turn greatly reduces computational effort at a corresponding loss of sampling efficiency. We further reduce the cost of forming the Fisher information matrix by using a low rank approximation via a randomized singular value decomposition technique. This is efficient since a small number of Hessian-vector products are required. The Hessian-vector product in turn requires only two extra PDE solves using the adjoint technique. Various numerical results up to 1025 parameters are presented to demonstrate the ability of the RMHMC method in exploring the geometric structure of the problem to propose (almost) uncorrelated/independent samples that are far away from each other, and yet the acceptance rate is almost unity. The results also suggest that for the PDE models considered the proposed fixed metric RMHMC can attain almost as high a quality performance as the original RMHMC, i.e. generating (almost) uncorrelated/independent samples, while being two orders of magnitude less computationally expensive.
The Bayesian approach to inverse problems typically relies on posterior
sampling approaches, such as Markov chain Monte Carlo, for which the generation
of each sample requires one or more evaluations of the parameter-to-observable
map or forward model. When these evaluations are computationally intensive,
approximations of the forward model are essential to accelerating sample-based
inference. Yet the construction of globally accurate approximations for
nonlinear forward models can be computationally prohibitive and in fact
unnecessary, as the posterior distribution typically concentrates on a small
fraction of the support of the prior distribution. We present a new approach
that uses stochastic optimization to construct polynomial approximations over a
sequence of measures adaptively determined from the data, eventually
concentrating on the posterior distribution. The approach yields substantial
gains in efficiency and accuracy over prior-based surrogates, as demonstrated
via application to inverse problems in partial differential equations.
This article is concerned with the propagation-of-chaos properties of genetic type particle models. This class of models arises in a variety of scientific disciplines including theoretical physics, macromolecular biology, engineering sciences, and more particularly in computational statis- tics and advanced signal processing. From the pure mathematical point of view, these interacting particle systems can be regarded as a mean field particle interpretation of a class of Feynman-Kac measures on path spaces. In the present article, we design an original integration theory of propagation-of-chaos based on the fluctuation analysis of a class of interacting particle random fields. We provide analytic functional representations of the distributions of finite particle blocks, yielding what seems to be the first result of this kind for interacting particle systems. These asymptotic expansions are expressed in terms of the limiting Feynman-Kac semigroups and a class of interacting jump operators. These results provide both sharp estimates of the negligible bias introduced by the interaction mechanisms, and central limit theorems for nondegenerate U-statistics and von Mises statistics associated with genealogical tree models. Applications to nonlinear filtering problems and interacting Markov chain Monte Carlo algorithms are discussed. 1. Introduction. In this article, we present an original fluctuation analysis of the propagation of chaos properties of a class of genetic type interacting particle sys- tems. This subject has various natural links to statistical physics, macromolecular biology, engineering sciences, and more particularly, to rare event estimation, nonlin- ear filtering, global optimization, and sequential Monte Carlo theory.
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 an asymptotic theory for a class of
such 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 contexts.
This paper discusses a novel strategy for simulating rare events and an associated Monte Carlo estimation of tail probabilities.
Our method uses a system of interacting particles and exploits a Feynman-Kac representation of that system to analyze their
fluctuations. Our precise analysis of the variance of a standard multilevel splitting algorithm reveals an opportunity for
improvement. This leads to a novel method that relies on adaptive levels and produces, in the limit of an idealized version
of the algorithm, estimates with optimal variance. The motivation for this theoretical work comes from problems occurring
in watermarking and fingerprinting of digital contents, which represents a new field of applications of rare event simulation
techniques. Some numerical results show performance close to the idealized version of our technique for these practical applications.
KeywordsRare event–Sequential importance sampling–Feynman-Kac formula–Metropolis-Hastings–Fingerprinting–Watermarking
Many problems arising in applications result in the need to probe a
probability distribution for functions. Examples include Bayesian nonparametric
statistics and conditioned diffusion processes. Standard MCMC algorithms
typically become arbitrarily slow under the mesh refinement dictated by
nonparametric description of the unknown function. We describe an approach to
modifying a whole range of MCMC methods, applicable whenever the target measure
has density with respect to a Gaussian process or Gaussian random field
reference measure, which ensures that their speed of convergence is robust
under mesh refinement. Gaussian processes or random fields are fields whose
marginal distributions, when evaluated at any finite set of N points, are
-valued Gaussians. The algorithmic approach that we describe is
applicable not only when the desired probability measure has density with
respect to a Gaussian process or Gaussian random field reference measure, but
also to some useful non-Gaussian reference measures constructed through random
truncation. In the applications of interest the data is often sparse and the
prior specification is an essential part of the overall modelling strategy.
These Gaussian-based reference measures are a very flexible modelling tool,
finding wide-ranging application. Examples are shown in density estimation,
data assimilation in fluid mechanics, subsurface geophysics and image
registration. The key design principle is to formulate the MCMC method so that
it is, in principle, applicable for functions; this may be achieved by use of
proposals based on carefully chosen time-discretizations of stochastic
dynamical systems which exactly preserve the Gaussian reference measure. Taking
this approach leads to many new algorithms which can be implemented via minor
modification of existing algorithms, yet which show enormous speed-up on a wide
range of applied problems.
We investigate the stability of a Sequential Monte Carlo (SMC) method applied
to the problem of sampling from a target distribution on 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 as with
. The convergence is achieved with a computational cost
proportional to . If , 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 as
and . Also, we show
that the Monte Carlo error for estimating a fixed dimensional marginal
expectation is of order 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.
We present an adaptive method for the automatic scaling of Random-Walk Metropolis-Hastings algorithms, which quickly and robustly identifies the scaling factor that yields a specified overall sampler acceptance probability. Our method relies on the use of the Robbins-Monro search process, whose performance is determined by an unknown steplength constant. We give a very simple estimator of this constant for proposal distributions that are univariate or multivariate normal, together with a sampling algorithm for automating the method. The effectiveness of the algorithm is demonstrated with both simulated and real data examples. This approach could be implemented as a useful component in more complex adaptive Markov chain Monte Carlo algorithms, or as part of automated software packages.
This contribution is devoted to the comparison of various resampling approaches that have been proposed in the literature on particle filtering. It is first shown using simple arguments that the so-called residual and stratified methods do yield an improvement over the basic multinomial resampling approach. A simple counter-example showing that this property does not hold true for systematic resampling is given. Finally, some results on the large-sample behavior of the simple bootstrap filter algorithm are given. In particular, a central limit theorem is established for the case where resampling is performed using the residual approach.
This book provides a basic introduction to reduced basis (RB) methods for problems involving the repeated solution of partial differential equations (PDEs) arising from engineering and applied sciences, such as PDEs depending on several parameters and PDE-constrained optimization. The book presents a general mathematical formulation of RB methods, analyzes their fundamental theoretical properties, discusses the related algorithmic and implementation aspects, and highlights their built-in algebraic and geometric structures. More specifically, the authors discuss alternative strategies for constructing accurate RB spaces using greedy algorithms and proper orthogonal decomposition techniques, investigate their approximation properties and analyze offline-online decomposition strategies aimed at the reduction of computational complexity. Furthermore, they carry out both a priori and a posteriori error analysis. The whole mathematical presentation is made more stimulating by the use of representative examples of applicative interest in the context of both linear and nonlinear PDEs. Moreover, the inclusion of many pseudocodes allows the reader to easily implement the algorithms illustrated throughout the text. The book will be ideal for upper undergraduate students and, more generally, people interested in scientific computing. © Springer International Publishing Switzerland 2016. All rights reserved.
We consider the problem of estimating the volume of the excursion
set of a function above a
given threshold, under a given probability measure on . In this
article, we combine the popular subset simulation algorithm (Au and Beck,
Probab. Eng. Mech. 2001) and our sequential Bayesian approach for the
estimation of a probability of failure (Bect, Ginsbourger, Li, Picheny and
Vazquez, Stat. Comput. 2012). This makes it possible to estimate when
the number of evaluations of f is very limited and is very small.
The resulting algorithm is called Bayesian subset simulation (BSS). A key idea,
as in the subset simulation algorithm, is to estimate the probabilities of a
sequence of excursion sets of f above intermediate thresholds, using a
sequential Monte Carlo (SMC) approach. A Gaussian process prior on f is used
to define the sequence of densities targeted by the SMC algorithm, and drive
the selection of evaluation points of f to estimate the intermediate
probabilities. Adaptive procedures are proposed to determine the intermediate
thresholds and the number of evaluations to be carried out at each stage of the
algorithm. Numerical experiments illustrate that BSS achieves significant
savings in the number of function evaluations with respect to other Monte~Carlo
approaches.
The goal of importance sampling is to estimate the expected value of a given
function with respect to a probability measure using a random sample of
size n drawn from a different probability measure . If the two measures
and are nearly singular with respect to each other, which is often
the case in practice, the sample size required for accurate estimation is
large. In this article it is shown that in a fairly general setting, a sample
of size approximately is necessary and sufficient for
accurate estimation by importance sampling, where is the
Kullback--Leibler divergence of from . In particular, the required
sample size exhibits a kind of cut-off in the logarithmic scale. The theory is
applied to obtain a fairly general formula for the sample size required in
importance sampling for exponential families (Gibbs measures). We also show
that the standard variance-based diagnostic for convergence of importance
sampling is fundamentally problematic. An alternative diagnostic that provably
works in certain situations is suggested.
Reduced basis methods are projection-based model order reduction techniques
for reducing the computational complexity of solving parametrized partial
differential equation problems. In this work we discuss the design of pyMOR, a
freely available software library of model order reduction algorithms, in
particular reduced basis methods, implemented with the Python programming
language. As its main design feature, all reduction algorithms in pyMOR are
implemented generically via operations on well-defined vector array, operator
and discretization interface classes. This allows for an easy integration with
existing third-party high-performance partial differential equation solvers
without adding any model reduction specific code to these solvers. Besides an
in-depth discussion of pyMOR's design philosophy and architecture, we present
several benchmark results and numerical examples showing the feasibility of our
approach.
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 . 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 . 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.
In this article we consider a Bayesian inverse problem associated to elliptic
partial differential equations (PDEs) in two and three dimensions. This class
of inverse problems is important in applications such as hydrology, but the
complexity of the link function between unknown field and measurements can make
it difficult to draw inference from the associated posterior. We prove that for
this inverse problem a basic SMC method has a Monte Carlo rate of convergence
with constants which are independent of the dimension of the discretization of
the problem; indeed convergence of the SMC method is established in a function
space setting. We also develop an enhancement of the sequential Monte Carlo
(SMC) methods for inverse problems which were introduced in \cite{kantas}; the
enhancement is designed to deal with the additional complexity of this elliptic
inverse problem. The efficacy of the methodology, and its desirable theoretical
properties, are demonstrated on numerical examples in both two and three
dimensions.
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.
Computing (ratios of) normalizing constants of probability models is a fundamental computational problem for many statistical and scientific studies. Monte Carlo simulation is an effective technique, especially with complex and high-dimensional models. This paper aims to bring to the attention of general statistical audiences of some effective methods originating from theoretical physics and at the same time to explore these methods from a more statistical perspective, through establishing theoretical connections and illustrating their uses with statistical problems. We show that the acceptance ratio method and thermodynamic integration are natural generalizations of importance sampling, which is most familiar to statistical audiences. The former generalizes importance sampling through the use of a single "bridge" density and is thus a case of bridge sampling in the sense of Meng and Wong. Thermodynamic integration, which is also known in the numerical analysis literature as Ogata's method for high-dimensional integration, corresponds to the use of infinitely many and continuously connected bridges (and thus a "path"). Our path sampling formulation offers more flexibility and thus potential efficiency to thermodynamic integration, and the search of optimal paths turns out to have close connections with the Jeffreys prior density and the Rao and Hellinger distances between two densities. We provide an informative theoretical example as well as two empirical examples (involving 17- to 70-dimensional integrations) to illustrate the potential and implementation of path sampling. We also discuss some open problems.
We propose a methodology to sample sequentially from a sequence of probability distributions that are defined on a common space, each distribution being known up to a normalizing constant. These probability distributions are approximated by a cloud of weighted random samples which are propagated over time by using sequential Monte Carlo methods. This methodology allows us to derive simple algorithms to make parallel Markov chain Monte Carlo algorithms interact to perform global optimization and sequential Bayesian estimation and to compute ratios of normalizing constants. We illustrate these algorithms for various integration tasks arising in the context of Bayesian inference. Copyright 2006 Royal Statistical Society.
Entropy minimizing distributions are worst-case optimal importance proposals
- Frédéric Cérou
- Patrick Héas
- Mathias Rousset
Frédéric Cérou, Patrick Héas, and Mathias Rousset. Entropy minimizing distributions are
worst-case optimal importance proposals. arXiv preprint arXiv:2212.04292, 2022.
Adaptive reduced multilevel splitting
- Frédéric Cérou
- Patrick Héas
- Mathias Rousset
Frédéric Cérou, Patrick Héas, and Mathias Rousset. Adaptive reduced multilevel splitting.
arXiv preprint arXiv:2312.15256, 2024.