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

This paper aims at illustrating the importance of using S-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 S-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 S-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 S-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.