Usage of a pair of -paths in Bayesian estimation of a unimodal density

This paper aims at illustrating the importance of using -paths in Bayesian estimation of a unimodal density on the real line. A class of species sampling mixture models containing random densities that are unimodal and not necessarily symmetric is considered. A novel and explicit characterization of the posterior distribution expressible as a finite mixture over pairs of two dependent -paths is derived, resulting in closed-form and tractable Bayes estimators for both the density and the mode as finite sums over the pairs. These results are statistically important as they are proved to be Rao-Blackwell improvements over existing results expressible in terms of partitions, and thus can be estimated with less variability. Extending an effective and newly-developed sequential importance sampling (SIS) scheme for sampling one -path at a time, an SIS scheme is proposed to approximate the density estimates or any other posterior quantities of the model that are expressible in terms of two -paths. Simulation results are reported to demonstrate practicality of our methodology and its effectiveness over an existing class of non-iterative algorithms that are based on sampling partitions. Indeed, the latter commonly-used algorithms, widely believed to be feasible, are shown to be ineffective and unreliable, implying that there exists hardly any practical non-iterative algorithm in this context. This prompts the essentiality of a practically useful algorithm for the problem.