Exponential model selection (in NMR) using Bayesian probability theory

In a companion article in this issue, parameter estimation using exponential models was addressed when the form of the model is known (i.e., when the number of exponentials and whether a constant offset is present are known). In this article, we apply Bayesian probability theory to the problem of determining the functional form of the model. The calculations are implemented using Markov chain Monte Carlo with simulated annealing to draw samples from the joint posterior probability for the parameters and the functional form of the model. Monte Carlo integration is then used to approximate the marginal posterior probabilities for all the parameters, including the number of exponentials and whether a constant offset is present. Examples using empirical data are given to illustrate the calculations.