Bayesian Inference With Incomplete Multinomial Data: A Problem in Pathogen Diversity

With recent advances in genetic analysis, it has become feasible to classify a pathogen into genetically distinct variants even though they apparently cause an infected subject similar symptoms. The availability of such data opens up the interesting problem of studying the spatiotemporal variation in the diversity of variants of a pathogen. Data on pathogen variants often suffer the problems of (i) low cell counts, (ii) incomplete classification due to laboratory problems (e.g., contamination), and (iii) unseen variants. Shannon's entropy may be used as a measure of variant diversity. A Bayesian approach can be used to deal with the problems of low cell counts and unseen variants. Bayesian analysis of incomplete multinomial data may be carried out by Markov chain Monte Carlo techniques. However, for pathogen-variant data, there often is only one source of missingness-namely, some subjects are known to be infected by some unidentified pathogen variant. We point out that for incomplete data with disjoint sources of missingness, Bayesian analysis can be done more efficiently using an iid sampling scheme from the posterior distribution. We illustrate the method by analyzing a data set on the prevalence of bartonella infection among individual colonies of prairie dogs at the study site in Colorado between 2003 and 2006. We compare the result from the proposed Monte Carlo method with the results from other methods, including a model that entertains within-variant spatial correlation but no between-variant spatial correlation. This article has supplementary material online.