Bayesian and least-squares approaches for estimating parameters of decay processes

Abstract

Bayesian and least squares approaches are applied to investigate further a simple, yet relevant, nonlinear example, namely, the estimation of the decay time of an exponential decay process with background. Though classical and Bayesian statistics interpret uncertainty claims in different ways, the results for both approaches are numerically compared here. The data were assumed to be independently and normally distributed with constant noise variance. Data sets are constructed under the exponential decay model with signal-to-noise ratios that differ by one and two orders of magnitude. Probability distributions for the decay time are calculated for two rectangular prior distributions with null lower bound whose upper bounds differ by two orders of magnitude. Specifically, two Bayesian estimates are compared with the least squares estimate: the posterior mode and the posterior mean. The roles and advantages of each estimate are discussed. The Bayesian credibility interval is also compared with the classical confidence interval. All the above calculation is performed with simple methods that can be easily implemented with those commercial laboratory software packages routinely used by measurement personnel for controlling instrumentation and reporting measurement results. Such methods are appropriate when the posterior can be calculated analytically as in the present case. The results of this investigation serve as subsidy to answer some questions posed in the article, namely: do Bayesian summaries need to be robust under changes in the prior? Are there any Bayesian summaries robust under changes in the observed data at repeated trials? If so, under what circumstances, and why? The amount of information provided by each experiment is also computed as it plays a key role in answering such questions.