The widely applicable Bayesian information criterion (WBIC) is a simple and fast approximation to the model evidence that has received little practical consideration. WBIC uses the fact that the log evidence can be written as an expectation, with respect to a powered posterior proportional to the likelihood raised to a power t * ∈ (0, 1), of the log deviance. Finding this temperature value t * is generally an intractable problem. We find that for a particular tractable statistical model that the mean squared error of an optimally-tuned version of WBIC with correct temperature t * is lower than an optimally-tuned version of thermodynamic integration (power posteriors). However in practice WBIC uses the a canonical choice of t = 1/ log(n). Here we investigate the performance of WBIC in practice, for a range of statistical models, both regular models and singular models such as latent variable models or those with a hierarchical structure for which BIC cannot provide an adequate solution. Our findings are that, generally WBIC performs adequately when one uses informative priors, but it can systematically overestimate the evidence, particularly for small sample sizes. cable Bayesian information criterion.where f (y|θ k , m k ) is the likelihood of the data under model m k with parameters θ k and p(θ k |m k ) is the prior on the parameters in model m k .The constant of proportionality for the un-normalised posterior distribution for model m k is the marginal likelihood or evidence,This is a vital quantity in Bayesian model choice and developing good estimates of it continues to be an active area of research in computational statistics. Henceforth, for brevity of notation, we will drop the dependence on m k , so that we refer to the evidence, likelihood and prior distribution for a given model as, p(y), f (y|θ), p(θ), respectively.There are a growing number of techniques to evaluate the evidence, see for instance, Gelman and Meng (1998) for a thorough review of importance, bridge and path sampling methods, Robert and Wraith (2009) for an updated review of such methods that includes the more recent mixture bridge-sampling approach (Chopin and Robert 2007), the generalised harmonic mean estimator (Gelfand and Dey 1994) and nested sampling ((Skilling 2006), or perhaps, (Burrows 1980)), in addition to (Friel and Wyse 2012) who compare the accuracy and computational burden of these methods.The contribution of this work is to explore a new method of approximating the evidence, the widely applicable Bayesian information criterion (WBIC) of Watanabe (2013). WBIC was motivated by the fact that the Bayesian information criterion (BIC or Schwarz criterion) (Schwarz 1978) is not applicable to singular models. A statistical model is termed regular if the mapping from model parameters to a probability distribution is one-to-one and if its Fisher information matrix is positive definite. Otherwise, a statistical model is singular. Singular models arise, for example, in latent variable models such as mixture models, hidden Markov model...

Max-stable processes are a common choice for modelling spatial extreme data as they arise naturally as the infinite-dimensional generalisation of multivariate extreme value theory. Statistical inference for such models is complicated by the intractability of the multivariate density function. Nonparametric, composite likelihood-based, and Bayesian approaches have been proposed to address this difficulty. More recently, a simulationbased approach using approximate Bayesian computation (ABC) has been employed for estimating parameters of max-stable models. ABC algorithms rely on the evaluation of discrepancies between model simulations and the observed data rather than explicit evaluations of computationally expensive or intractable likelihood functions. The use of an ABC method to perform model selection for max-stable models is explored. Three max-stable models are regarded: the extremal-t model with either a Whittle-Matérn or a powered exponential covariance function, and the Brown-Resnick model with power variogram. In addition, the non-extremal Student-t copula model with a Whittle-Matérn or a powered exponential covariance function is also considered. The method is applied to annual maximum temperature data from 25 weather stations dispersed around South Australia.

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