191 research outputs found
Unbiased and Consistent Nested Sampling via Sequential Monte Carlo
We introduce a new class of sequential Monte Carlo methods called Nested
Sampling via Sequential Monte Carlo (NS-SMC), which reframes the Nested
Sampling method of Skilling (2006) in terms of sequential Monte Carlo
techniques. This new framework allows convergence results to be obtained in the
setting when Markov chain Monte Carlo (MCMC) is used to produce new samples. An
additional benefit is that marginal likelihood estimates are unbiased. In
contrast to NS, the analysis of NS-SMC does not require the (unrealistic)
assumption that the simulated samples be independent. As the original NS
algorithm is a special case of NS-SMC, this provides insights as to why NS
seems to produce accurate estimates despite a typical violation of its
assumptions. For applications of NS-SMC, we give advice on tuning MCMC kernels
in an automated manner via a preliminary pilot run, and present a new method
for appropriately choosing the number of MCMC repeats at each iteration.
Finally, a numerical study is conducted where the performance of NS-SMC and
temperature-annealed SMC is compared on several challenging and realistic
problems. MATLAB code for our experiments is made available at
https://github.com/LeahPrice/SMC-NS .Comment: 45 pages, some minor typographical errors fixed since last versio
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 bound on the
cost to obtain a mean-square error of . 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
A Multilevel Approach for Stochastic Nonlinear Optimal Control
We consider a class of finite time horizon nonlinear stochastic optimal
control problem, where the control acts additively on the dynamics and the
control cost is quadratic. This framework is flexible and has found
applications in many domains. Although the optimal control admits a path
integral representation for this class of control problems, efficient
computation of the associated path integrals remains a challenging Monte Carlo
task. The focus of this article is to propose a new Monte Carlo approach that
significantly improves upon existing methodology. Our proposed methodology
first tackles the issue of exponential growth in variance with the time horizon
by casting optimal control estimation as a smoothing problem for a state space
model associated with the control problem, and applying smoothing algorithms
based on particle Markov chain Monte Carlo. To further reduce computational
cost, we then develop a multilevel Monte Carlo method which allows us to obtain
an estimator of the optimal control with mean squared
error with a computational cost of
. In contrast, a computational cost
of is required for existing methodology to achieve
the same mean squared error. Our approach is illustrated on two numerical
examples, which validate our theory
An invitation to sequential Monte Carlo samplers
Sequential Monte Carlo samplers provide consistent approximations of
sequences of probability distributions and of their normalizing constants, via
particles obtained with a combination of importance weights and Markov
transitions. This article presents this class of methods and a number of recent
advances, with the goal of helping statisticians assess the applicability and
usefulness of these methods for their purposes. Our presentation emphasizes the
role of bridging distributions for computational and statistical purposes.
Numerical experiments are provided on simple settings such as multivariate
Normals, logistic regression and a basic susceptible-infected-recovered model,
illustrating the impact of the dimension, the ability to perform inference
sequentially and the estimation of normalizing constants.Comment: review article, 34 pages, 10 figure
Estimation and uncertainty quantification for the output from quantum simulators
The problem of estimating certain distributions over is
considered here. The distribution represents a quantum system of qubits,
where there are non-trivial dependencies between the qubits. A maximum entropy
approach is adopted to reconstruct the distribution from exact moments or
observed empirical moments. The Robbins Monro algorithm is used to solve the
intractable maximum entropy problem, by constructing an unbiased estimator of
the un-normalized target with a sequential Monte Carlo sampler at each
iteration. In the case of empirical moments, this coincides with a maximum
likelihood estimator. A Bayesian formulation is also considered in order to
quantify posterior uncertainty. Several approaches are proposed in order to
tackle this challenging problem, based on recently developed methodologies. In
particular, unbiased estimators of the gradient of the log posterior are
constructed and used within a provably convergent Langevin-based Markov chain
Monte Carlo method. The methods are illustrated on classically simulated output
from quantum simulators
A randomized Multi-index sequential Monte Carlo method
We consider the problem of estimating expectations with respect to a target
distribution with an unknown normalizing constant, and where even the
unnormalized target needs to be approximated at finite resolution. Under such
an assumption, this work builds upon a recently introduced multi-index
Sequential Monte Carlo (SMC) ratio estimator, which provably enjoys the
complexity improvements of multi-index Monte Carlo (MIMC) and the efficiency of
SMC for inference. The present work leverages a randomization strategy to
remove bias entirely, which simplifies estimation substantially, particularly
in the MIMC context, where the choice of index set is otherwise important.
Under reasonable assumptions, the proposed method provably achieves the same
canonical complexity of MSE^(-1) as the original method, but without
discretization bias. It is illustrated on examples of Bayesian inverse
problems.Comment: 26 pages 6 figure
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