Generalized species sampling priors with latent Beta reinforcements

Abstract

Many popular Bayesian nonparametric priors can be characterized in terms of exchangeable species sampling sequences. However, in some applications, exchangeability may not be appropriate. We introduce a novel and probabilistically coherent family of non-exchangeable species sampling sequences characterized by a tractable predictive probability function with weights driven by a sequence of independent Beta random variables. We compare their theoretical clustering properties with those of the Dirichlet Process and the two parameters Poisson-Dirichlet process. The proposed construction provides a complete characterization of the joint process, differently from existing work. We then propose the use of such process as prior distribution in a hierarchical Bayes modeling framework, and we describe a Markov Chain Monte Carlo sampler for posterior inference. We evaluate the performance of the prior and the robustness of the resulting inference in a simulation study, providing a comparison with popular Dirichlet Processes mixtures and Hidden Markov Models. Finally, we develop an application to the detection of chromosomal aberrations in breast cancer by leveraging array CGH data.

Keywords: Bayesian non-parametrics; Cancer; Genomics; MCMC; Predictive Probability Functions; Random Partitions; Species Sampling Priors.

Figure 1

Posterior distribution of the number of clusters in the simulation of Section 5.3 (τ = 0.25). Case (a) corresponds to a Beta-GOS(α_n = n, β_n = 1), case (b) to a Beta-GOS(α_n = β_n = 1) and case (c) to a Dirichlet Process with parameter θ = 1.

Figure 2

Illustrative segmentation-type plots for the simulation study in Section 5.4. Column (a): subset of data for two replicates. Column (b) top: an example of allocation for a Beta-Gos(α_n = 1, β_n = 1) plotted vs the truth (black line); column (b) bottom considers a Beta-Gos(α_n = n, β_n = 1). Column (c) illustrates the fitting by a HMM with 4 states.

Figure 3

Model fit overview: Array CGH gains and losses on chromosome 8 for two samples of breast tumors in the dataset in (Chin et al., 2006). Points with different shapes denote different clusters.

Figure 4

A) Frequencies of genome copy number gains and losses plotted as a function of genomic location. B) Frequency of tumors showing high-level amplification. The dashed vertical lines separate the 23 chromosomes.