Bayes estimates of multimodal density features using DNA and Economic Data

In several scientific fields, like bioinformatics, financial and macro-economics, important theoretical and practical issues exist that involve multimodal data distributions. We propose a Bayesian approach using mixtures distributions to approximate accurately such data distributions. Shape and other features of the mixture approximations are estimated including their uncertainty. For discrete data, we introduce a novel mixture of shifted Poisson distributions with an unknown number of components, which overcomes the equidispersion restriction in the standard Poisson which accomodates a wide range of shapes such as multimodality and long tails. Our simulation-based Bayesian inference treats the density features as random variables and highest credibility regions around features are easily obtained. For discrete data we develop an adapted version of the Reversible Jump Markov Chain Monte Carlo (RJMCMC) method, which allows for an unknown number of components instead of the more restrictive approach of choosing a particular number of mixture components using information criteria. Using simulated data, we show that our approach works successfully for three issues that one encounters during the estimation of mixtures: label switching; mixture complexity and prior information and mode membership versus component membership. The proposed method is applied to three empirical data sets: The count data method yields a novel perspective of the data on DNA tandem repeats in \cite{DNA_leiden}; the bimodal distribution of payment details of clients obtaining a loan from a financial institution in Spain in 1990 gives insight into the repayment ability of individual clients; and the distribution of the modes of real GDP growth data from the PennWorld Tables and their evolution over time explores possible world-wide economic convergence as well as group convergence between the US and European countries. The results of our descriptive analysis may be used as input for forecasting and policy analysis.