Bayesian Inference and Ill-Posed Inverse Problems

David Schmidt (P-21)


Ill-posed inverse problems, in which there are many different solutions that could have produced the given data, are common. For example, think about the job your audio CD player has. Bayesian inference is well suited to such problems because it provides a well-defined way to incorporate prior information that can reduce the ambiguity or range of likely solutions. A posterior probability distribution over the space of possible solutions is the result, which encapsulates all the information available and can be used to make probabilistic inferences. Although prior information is sometimes considered subjective and its use may be controversial, prior information is essential for ill-posed inverse problem, in order to reduce the range of likely solutions. Indeed, one should tailor the prior distribution to incorporate all of the pertinent prior information available for each problem in order to maximize the specificity of the resulting posterior distribution. Even so, the posterior distribution may be broad and the most likely solution may not be representative of the full range of likely solutions. In such cases it is important to consider the full range of likely solutions when making inferences. Examples of the use of Bayesian inference will be presented for a problem in human brain mapping and for the problem of inferring the continuous distribution from which a finite data sample have been drawn. Finally, I will discuss implications of applying this approach to problems in complex modeling and simulation.