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We introduce the weighted likelihood bootstrap (WLB) as a way to simulate approximately from a posterior distribution. This method is often easy to implement, requiring only an algorithm for calculating the maximum likelihood estimator, such as iteratively reweighted least squares. In the generic weighting scheme, the WLB is first order correct under quite general conditions. Inaccuracies can be removed by using the WLB as a source of samples in the sampling-importance resampling (SIR) algorithm, which also allows incorporation of particular prior information. The SIR-adjusted WLB can be a competitive alternative to other integration methods in certain models. Asymptotic expansions elucidate the second-order properties of the WLB, which is a generalization of Rubin's Bayesian bootstrap. The calculation of approximate Bayes factors for model comparison is also considered. We note that, given a sample simulated from the posterior distribution, the required marginal likelihood may be simulation consistently estimated by the harmonic mean of the associated likelihood values; a modification of this estimator that avoids instability is also noted. These methods provide simple ways of calculating approximate Bayes factors and posterior model probabilities for a very wide class of models.

SUMMARY We state and prove a simple and general expression for the marginal mean in an exponential family state space model with conjugate state density. We then discuss implications of this result on the choice of parameterization of the observation density, on the interpretation of the predictive density for steady models and on exponentially weighted moving average and trend-free forecasting.

On introduit un modele pour des chaines de Markov du leme ordre qui comporte seulement un parametre additionnel pour chaque retard

On developpe une approche bayesienne au probleme de la comparaison de modeles pour des processus de Poisson non homogenes

Spiegelhalter and Smith (1982)hereafter SSproposed an approximate method for calculating the Bayes factor for a log-linear model Mo for a contingency table against the saturated model M1, with vague prior information. We adopt their notation and denote by SS(n) equation (n) of SS. Suppose x1, ..., Xk have a multinomial distribution with parameters 014 ..., 4k where Xi 0 0, X4i = 1. We write yT = (log x1, ..., log xk), 01 = (log 01, ..., log dk), and Y = diag{x1, ..., Xk}. Then if M1 is the saturated model, and Mo is the nested, log-linear, model defined by setting the contrasts CO1 = 0, where C is an s x k matrix with rank s and rows summing to zero, the approximate Bayes factor Bo1 for Mo against M1 is given by SS (32). This, however, is indeterminate if any of the cell frequencies in the table is zero. This is because of the use by SS of a prior density proportional to (H+i)'. If, however, we use, instead, the standard Jeffreys prior density proportional to (I1i) 1/2, the problem no longer arises. Then, by the arguments of SS and Lindley (1964), the resulting Bayes factor is still given by SS (32), with xi replaced by xi + 2 in the definitions of y and Y (i = 1, . . ., k), and SS (33) replaced by c 1 = (2 )s2 1 CCT I 1/2. If this solution is adopted, the prior is proper, and so, in principle, the problem of assigning an arbitrary multiplicative constant, for which the SS approach was primarily devised, need not arise. One could, in theory, simply apply Bayes; theorem directly and so obtain the Bayes factor exactly. However, in practice, this is difficult to do, and I know of no general solution to the problem. Even for simpler, more specific, contingency table and related models, such as those of independence or equiprobability, finding the Bayes factor exactly is not too easy; see, for example, Crook and Good (1982), Altham (1971), Gunel and Dickey (1974), Gunel (1982), Broniatowski (1981), and references therein. The second purpose of this note is to point out that, conditionally on MO