6,248 research outputs found
Asymptotic Properties of Approximate Bayesian Computation
Approximate Bayesian computation allows for statistical analysis in models
with intractable likelihoods. In this paper we consider the asymptotic
behaviour of the posterior distribution obtained by this method. We give
general results on the rate at which the posterior distribution concentrates on
sets containing the true parameter, its limiting shape, and the asymptotic
distribution of the posterior mean. These results hold under given rates for
the tolerance used within the method, mild regularity conditions on the summary
statistics, and a condition linked to identification of the true parameters.
Implications for practitioners are discussed.Comment: This 31 pages paper is a revised version of the paper, including
supplementary materia
Regression approaches for Approximate Bayesian Computation
This book chapter introduces regression approaches and regression adjustment
for Approximate Bayesian Computation (ABC). Regression adjustment adjusts
parameter values after rejection sampling in order to account for the imperfect
match between simulations and observations. Imperfect match between simulations
and observations can be more pronounced when there are many summary statistics,
a phenomenon coined as the curse of dimensionality. Because of this imperfect
match, credibility intervals obtained with regression approaches can be
inflated compared to true credibility intervals. The chapter presents the main
concepts underlying regression adjustment. A theorem that compares theoretical
properties of posterior distributions obtained with and without regression
adjustment is presented. Last, a practical application of regression adjustment
in population genetics shows that regression adjustment shrinks posterior
distributions compared to rejection approaches, which is a solution to avoid
inflated credibility intervals.Comment: Book chapter, published in Handbook of Approximate Bayesian
Computation 201
Efficient learning in Approximate Bayesian Computation
Efficient learning in Approximate Bayesian Computatio
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
Approximate Bayesian Computation with composite score functions
Both Approximate Bayesian Computation (ABC) and composite likelihood methods
are useful for Bayesian and frequentist inference, respectively, when the
likelihood function is intractable. We propose to use composite likelihood
score functions as summary statistics in ABC in order to obtain accurate
approximations to the posterior distribution. This is motivated by the use of
the score function of the full likelihood, and extended to general unbiased
estimating functions in complex models. Moreover, we show that if the composite
score is suitably standardised, the resulting ABC procedure is invariant to
reparameterisations and automatically adjusts the curvature of the composite
likelihood, and of the corresponding posterior distribution. The method is
illustrated through examples with simulated data, and an application to
modelling of spatial extreme rainfall data is discussed.Comment: Statistics and Computing (final version
Approximate Bayesian computation via the energy statistic
Approximate Bayesian computation (ABC) has become an essential part of the
Bayesian toolbox for addressing problems in which the likelihood is
prohibitively expensive or entirely unknown, making it intractable. ABC defines
a pseudo-posterior by comparing observed data with simulated data,
traditionally based on some summary statistics, the elicitation of which is
regarded as a key difficulty. Recently, using data discrepancy measures has
been proposed in order to bypass the construction of summary statistics. Here
we propose to use the importance-sampling ABC (IS-ABC) algorithm relying on the
so-called two-sample energy statistic. We establish a new asymptotic result for
the case where both the observed sample size and the simulated data sample size
increase to infinity, which highlights to what extent the data discrepancy
measure impacts the asymptotic pseudo-posterior. The result holds in the broad
setting of IS-ABC methodologies, thus generalizing previous results that have
been established only for rejection ABC algorithms. Furthermore, we propose a
consistent V-statistic estimator of the energy statistic, under which we show
that the large sample result holds, and prove that the rejection ABC algorithm,
based on the energy statistic, generates pseudo-posterior distributions that
achieves convergence to the correct limits, when implemented with rejection
thresholds that converge to zero, in the finite sample setting. Our proposed
energy statistic based ABC algorithm is demonstrated on a variety of models,
including a Gaussian mixture, a moving-average model of order two, a bivariate
beta and a multivariate -and- distribution. We find that our proposed
method compares well with alternative discrepancy measures.Comment: 25 pages, 6 figures, 5 table
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
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