Order-Restricted Bayesian Estimation of the Age Composition of a Population of Atlantic Cod

Abstract

The management of a commercial fishery often requires estimates of the age composition. These estimates are typically based on age and length data obtained from sampling the commercial landings from the fishery and the catches from a research vessel survey. We use data from annual research cruises of Georges Bank conducted by the Canadian Department of Fisheries and Oceans to show how inferences for Atlantic cod (Gadus morhua) can be improved. Traditionally, two-phase stratified sampling is used with fish length (or weight) as the stratification variable. Letting Pi . denote the proportion of fish belonging to length stratum i, and π ij denote the proportion of fish belonging to age class j in stratum i, we use Bayesian methods to estimate P.j = π i πij , the proportion of fish that are age j. Specifications of smoothness, expressed as unimodal order relations among the π ij (within and between the length strata), are incorporated into the prior distributions. Uncertainty about both the locations of the modes and the unimodality itself are included as part of the probabilistic specification. With computations facilitated by using the Gibbs sampler, we show that the smoothness conditions provide very large gains in precision. For the data analyzed in this article, one can obtain similar precision by using (a) a conventional analysis or (b) an analysis with order restrictions and a sample of half the size in (a). We also show that better estimates of the age composition provide improved estimates of the quantities used by fisheries managers to forecast the catches from cohorts of fish.