Sensitivity of Bayes estimators to hyper-parameters with an application to maximum yield from fisheries

Priors are seldom unequivocal and an important component of Bayesian modeling is assessment of the sensitivity of the posterior to the specified prior distribution. This is especially true in fisheries science where the Bayesian approach has been promoted as a rigorous method for including existing information from previous surveys and from related stocks or species. These informative priors may be highly contested by various interest groups. Here, formulae for the first and second derivatives of Bayes estimators with respect to hyper-parameters of the joint prior density are given. The formula for the second derivative provides a correction to a previously published result. The formulae are shown to reduce to very convenient and easily implemented forms when the hyper-parameters are for exponential family marginal priors. For model parameters with such priors it is shown that the ratio of posterior variance to prior variance can be interpreted as the sensitivity of the posterior mean to the prior mean. This methodology is applied to a nonlinear state-space model for the biomass of South Atlantic albacore tuna and sensitivity of the maximum sustainable yield to the prior specification is examined.