Exponential parameter estimation (in NMR) using Bayesian probability theory

Abstract

Data modeled as sums of exponentials arise in many areas of science and are common in NMR. However, exponential parameter estimation is fundamentally a difficult problem. In this article, Bayesian probability theory is used to obtain optimal exponential parameter estimates. The calculations are implemented using Markov chain Monte Carlo with simulated annealing to draw samples from the joint posterior probability for all of the parameters appearing in the exponential model. Monte Carlo integration is then used to approximate the marginal posterior probabilities for each of the parameters. We give numerical examples taken from simulated data and NMR relaxation experiments to illustrate the calculations and the effect of prior information on the parameter estimates.

Exponential model selection (in NMR) using Bayesian probability theory

“…To perform this calculation, the model must be part of the inference problem. Equations [1][2][3][4] are the four different functional forms considered, but those equations do not by themselves designate a model. For example, Eq.…”

“…[10] is the prior probability for the parameters given the model indicator times the direct probability for the data, given the parameters and the model indicator. This integrand is the posterior probability for the parameters given the model indicator, and the calculation of this probability was addressed in (1). The details of factoring and assigning these probabilities are not repeated here, and we use the results from (1).…”

“…In that article, the number of exponentials and whether a constant offset is present were assumed to be known. As demonstrated in (1), if the exponential model is not in one-to-one correspondence with the data, the parameter estimates may significantly disagree with the estimates obtained using the correct model. Consequently, it is important for the model to correspond exactly to the data.…”

“…In a companion article in this issue (1), the parameter estimation problem given an exponential model was addressed using Bayesian probability theory. In that article, the number of exponentials and whether a constant offset is present were assumed to be known.…”

“…A j exp͕Ϫ␣ j t i ͖ ϩ n i [1] where d i is the data item sampled at t i , C is the constant offset, A j and ␣ j are the amplitude and decay rate constant of the jth exponential, n i represents the noise or measurement error, and m is the number of exponentials. The second form is…”