Bayesian Spatial Inversion and Conjugate Selection Gaussian Prior Models

We introduce the concept of conjugate prior models for a given likelihood function in Bayesian spatial inversion. The conjugate class of prior models can be selection extended and still remain conjugate. We demonstrate the generality of selection Gaussian prior models, representing multi-modality, skewness and heavy-tailedness. For Gauss-linear likelihood functions, the posterior model is also selection Gaussian. The model parameters of the posterior pdf are explisite functions of the model parameters of the likelihood and prior models - and the actual observations, of course. Efficient algorithms for simulation of and prediction for the selection Gaussian posterior pdf are defined. Inference of the model parameters in the selection Gaussian prior pdf, based on one training image of the spatial variable, can be reliably made by a maximum likelihood criterion and numerical optimization. Lastly, a seismic inversion case study is presented, and improvements of $20$-$40\%$ in prediction mean-square-error, relative to traditional Gaussian inversion, are found.