# Bayesian Spatial Inversion and Conjugate Selection Gaussian Prior Models

## TL;DR

This paper develops conjugate prior models for Bayesian spatial inversion, introducing selection Gaussian priors that handle complex data features and demonstrate improved seismic inversion accuracy over traditional methods.

## Contribution

It introduces selection Gaussian prior models that extend conjugate priors, enabling more flexible modeling of spatial data with multi-modality, skewness, and heavy tails.

## Key findings

- Selection Gaussian priors are conjugate for Gauss-linear likelihoods.
- Posterior models remain selection Gaussian with explicit parameter relationships.
- Seismic inversion shows 20-40% improvement in mean-square-error.

## Abstract

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.