Bayesian inversion with spatial priors: an exact sampling alternative to McMC

Geoscientists often use discretized spatial models of the earth where data are available in each grid cell providing independent information about model parameters of interest at that location. In Bayesian inference this information is given as a set of likelihoods describing the (unnormalized) probability of the model parameters given the data in each cell. Information about parameter values/spatial-correlations is described by a prior. The prior, likelihoods and Bayes' rule are used to specify a posterior over all model parameters. Due to the high dimensionality of typical models, the posterior is usually only known up to a multiplicative constant. Markov-chain Monte-Carlo (McMC) methods are then used to produce an ensemble of correlated samples from the posterior. These ensembles may not converge in finite time and detecting their state of convergence is often impossible in practice. Thus estimates of the posterior obtained in this way may be biased. We describe an alternative recursive algorithm to sample exactly from the posterior, so avoiding these convergence issues. We propose approximations to the algorithm so it may be used on large 2D model grids. We apply the algorithm to synthetic seismic attribute data and obtain results which compare favorably to the results of McMC sampling.