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\footnotesize \noindent Statistica Sinica {\bf 22} (2012), 813-845

\footnotesize \noindent doi:http://dx.doi.org/10.5705/ss.2009.248

\vspace{3pc} \normalsize \centerline{\large\bf BAYESIAN INFERENCE
FROM COMPOSITE LIKELIHOODS,}\vspace{2pt} \centerline{\large\bf  WITH
AN
  APPLICATION TO SPATIAL EXTREMES}
\vspace{.25cm}
\author{Mathieu Ribatet, Daniel Cooley and Anthony C. Davison}
\vspace{.4cm} \centerline{\it Universit\'e Montpellier II, Colorado
State University} \centerline{\it and \'Ecole Polytechnique
F\'ed\'erale de
  Lausanne}
\vspace{.55cm}
\begin{quotation}
\noindent {\it Abstract:}
  Composite likelihoods are increasingly used in applications where
  the full likelihood is analytically unknown or computationally
  prohibitive. Although some frequentist properties of the maximum
  composite likelihood estimator are akin to those of the maximum
  likelihood estimator, Bayesian inference based on composite
  likelihoods is in its early stages. This paper discusses inference
  when one uses composite likelihood in Bayes' formula.  We establish
  that using a composite likelihood results in a proper posterior
  density, though it can differ considerably from that stemming from
  the full likelihood.
  Building on previous work on composite likelihood ratio tests, we
  use asymptotic theory for misspecified models to propose two
  adjustments to the composite likelihood to
  obtain
  appropriate inference.  We also investigate use of the Metropolis
  Hastings algorithm and two implementations of the Gibbs sampler for
  obtaining draws from the composite posterior.
  We test the methods on simulated data and apply them to a spatial
  extreme rainfall dataset.  For the simulated data, we find that
  posterior credible intervals yield appropriate empirical coverage
  rates.  For the extreme precipitation data, we are able to both
  effectively model marginal behavior throughout the study region and
  obtain appropriate measures of spatial dependence.\par
\vspace{9pt} \noindent {\it Key words and phrases:}
 Bayesian hierarchical model, composite
  likelihood, Gibbs sampler, Markov chain Monte Carlo, max-stable
  process, Metropolis--Hastings algorithm, posterior coverage,
  rainfall data.\par
\end{quotation}
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