Impacts of atypical data on Bayesian inference and robust Bayesian approach in fisheries

Bayesian inference is increasingly used in fisheries. In formulating likelihood functions in Bayesian inference, data have been analyzed as if they are normally, identically, and independently distributed. It has come to be believed that the first two of the assumptions are frequently inappropriate in fisheries studies. In fact, data distributions are likely to be leptokurtic and (or) contaminated by occasional bad values giving rise to outliers in many fisheries studies. Despite the likelihood of having outliers in fisheries studies, the impacts of outliers on Bayesian inference have received little attention. In this study, using a simple growth model as an example, we evaluate the impacts of outliers on the derivation of posterior distributions in Bayesian analyses. Posterior distributions derived from the Bayesian method commonly used in fisheries are found to be sensitive to outliers. The distributions are severely biased in the presence of atypical values. The sensitivity of normality-based Bayesian analyses on atypical data may result from small "tails" of normal distribution so that the probability of occurrence of an event drops off quickly as one moves away from the mean a distance of a few standard deviations. A robust Bayesian method can be derived by including a mixture distribution that increases the size of tail so that the probability of occurrence of an event does not drop off too quickly as one moves away from the mean. The posterior distributions derived from this proposed approach are found to be robust to atypical data in this study. The proposed approach offers a potentially useful addition to Bayesian methods used in fisheries.