Newton and Raftery (1994) [284] Proposes the `weighted likelihood boostrap' (a non-MCMC, importance sampling approach) for simulating from posterior distributions. Notes on how the sampled values (from this or MCMC) can be used to estimate p(D given M) and Bayes factors.
Chib (1995) [78] Estimating p(D given M) from Gibbs sampling output, when the full conditionals (including proportionality constants) are known. Essentially reduces to estimating the posterior density at some (arbitrary) paramater values.
Verdinelli and Wasserman (1995) [377] Estimating BF for nested hypothesis from a sample from the posterior, based on the `Savage-Dickey density ratio'.
Carlin and Chib (1995) [73] An MCMC approach for model choice, where the model indicator is treated as one of the unknowns (i.e. the sampler traverses the model space as well as the parameter space). Moves between parameter spaces of different models are achieved using a `linking density'.
Green (1995) [169] Proposes a `reversible jump' MCMC method which traverses the whole model and parameter space, including jumps between models (and their parameter spaces).
Lewis and Raftery (1997) [238]
Laplace-Metropolis estimator
of $p(y|M)$ in BF: (1) usual Laplace approximation; (2) estimate
posterior mode and inverse Hessian using output from an MCMC algorithm.
Modification for hierarchical models (many nuisance parameters).
Variance using batch variability. Good performance in numerical
examples.
Brown et al. (1998) [61] Variable selection for multivariate regression models (concentrating on the linear model). Model indicators as a latent variables, posterior distribution for them. Closed-form results, but MCMC still required if number of possible predictors is large.
DiCiccio et al. (1997) [110] Comapring and extending various methods of estimating p(D given M) using MCMC or other simulation output.
Albert and Chib (1997) [15] MCMC methods for estimating Bayes factors for hierarchical models (in particular for comparing a fixed-effect and random-effect model).