13,748 research outputs found
Simulation in Statistics
Simulation has become a standard tool in statistics because it may be the
only tool available for analysing some classes of probabilistic models. We
review in this paper simulation tools that have been specifically derived to
address statistical challenges and, in particular, recent advances in the areas
of adaptive Markov chain Monte Carlo (MCMC) algorithms, and approximate
Bayesian calculation (ABC) algorithms.Comment: Draft of an advanced tutorial paper for the Proceedings of the 2011
Winter Simulation Conferenc
Adaptive approximate Bayesian computation for complex models
Approximate Bayesian computation (ABC) is a family of computational
techniques in Bayesian statistics. These techniques allow to fi t a model to
data without relying on the computation of the model likelihood. They instead
require to simulate a large number of times the model to be fi tted. A number
of re finements to the original rejection-based ABC scheme have been proposed,
including the sequential improvement of posterior distributions. This technique
allows to de- crease the number of model simulations required, but it still
presents several shortcomings which are particu- larly problematic for costly
to simulate complex models. We here provide a new algorithm to perform adaptive
approximate Bayesian computation, which is shown to perform better on both a
toy example and a complex social model.Comment: 14 pages, 5 figure
Approximate Bayesian Computation by Subset Simulation
A new Approximate Bayesian Computation (ABC) algorithm for Bayesian updating
of model parameters is proposed in this paper, which combines the ABC
principles with the technique of Subset Simulation for efficient rare-event
simulation, first developed in S.K. Au and J.L. Beck [1]. It has been named
ABC- SubSim. The idea is to choose the nested decreasing sequence of regions in
Subset Simulation as the regions that correspond to increasingly closer
approximations of the actual data vector in observation space. The efficiency
of the algorithm is demonstrated in two examples that illustrate some of the
challenges faced in real-world applications of ABC. We show that the proposed
algorithm outperforms other recent sequential ABC algorithms in terms of
computational efficiency while achieving the same, or better, measure of ac-
curacy in the posterior distribution. We also show that ABC-SubSim readily
provides an estimate of the evidence (marginal likelihood) for posterior model
class assessment, as a by-product
Bayesian subset simulation
We consider the problem of estimating a probability of failure ,
defined as 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 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 above
intermediate thresholds, using a sequential Monte Carlo (SMC) approach. A
Gaussian process prior on is used to define the sequence of densities
targeted by the SMC algorithm, and drive the selection of evaluation points of
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
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
On computational tools for Bayesian data analysis
While Robert and Rousseau (2010) addressed the foundational aspects of
Bayesian analysis, the current chapter details its practical aspects through a
review of the computational methods available for approximating Bayesian
procedures. Recent innovations like Monte Carlo Markov chain, sequential Monte
Carlo methods and more recently Approximate Bayesian Computation techniques
have considerably increased the potential for Bayesian applications and they
have also opened new avenues for Bayesian inference, first and foremost
Bayesian model choice.Comment: This is a chapter for the book "Bayesian Methods and Expert
Elicitation" edited by Klaus Bocker, 23 pages, 9 figure
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