Generalized priors in Bayesian inversion problems

Inverse problems, when only hard data are considered, are mathematically ill-posed because they have multiple solutions and are sensitive to small changes in the data. An example is the determination of hydraulic conductivity from head data. Bayesian methods provide a general stochastic framework within which one can solve such problems. The approach allows one to combine hard data with other information in order to explore the range of possible solutions. We consider the role of generalized (also known as improper) probability distributions within this framework. These distributions are very useful because they represent information parsimoniously (with simple equations and few parameters) and are ideal in terms of representing situations with limited information. They are particularly useful in introducing prior information about the dependent variables of the inverse problem and structural parameters that describe the degree of continuity or smoothness of dependent variables. Additionally, they are very useful from a computational perspective because they can lead to formulations with special structure, such as with high degree of sparsity or structure that can be exploited through high performance computing algorithms. We examine in the context of a simple example some of the consequences of using different generalized priors.