Spatial mixture modelling for unobserved point processes: examples in immunofluorescence histology

Abstract

We discuss Bayesian modelling and computational methods in analysis of indirectly observed spatial point processes. The context involves noisy measurements on an underlying point process that provide indirect and noisy data on locations of point outcomes. We are interested in problems in which the spatial intensity function may be highly heterogenous, and so is modelled via flexible nonparametric Bayesian mixture models. Analysis aims to estimate the underlying intensity function and the abundance of realized but unobserved points. Our motivating applications involve immunological studies of multiple fluorescent intensity images in sections of lymphatic tissue where the point processes represent geographical configurations of cells. We are interested in estimating intensity functions and cell abundance for each of a series of such data sets to facilitate comparisons of outcomes at different times and with respect to differing experimental conditions. The analysis is heavily computational, utilizing recently introduced MCMC approaches for spatial point process mixtures and extending them to the broader new context here of unobserved outcomes. Further, our example applications are problems in which the individual objects of interest are not simply points, but rather small groups of pixels; this implies a need to work at an aggregate pixel region level and we develop the resulting novel methodology for this. Two examples with with immunofluorescence histology data demonstrate the models and computational methodology.

Hierarchical Bayesian mixture modelling for antigen-specific T-cell subtyping in combinatorially encoded flow cytometry studies

“…Further background can be found in, for example Ishwaran and James (2001) or the summary appendix in Ji et al (2009). First, π 1 = ν 1 , π j = (1− ν 1 )…(1− ν j −1 ) ν j , where ν j , j ≠ J ~ Be (1, α b ), ν J = 1, and the hyper-parameter α b ~ Ga ( e b , f b ) for some specified e b and f b . Second, independently of the π j the normal mean and variance matrices are independent across components with priors…”

“…Further background can be found in, for example Ishwaran and James (2001) or the summary appendix in Ji et al (2009).…”

“…Sampling mixture probabilities π 1: J and the hyperparameter of the DP model use a Metropolis-Hastings extension of the standard component distributions (Ishwaran and James, 2001; Ji et al, 2009), as follows. The mixture probabilities π j are obtained from underlying beta variates ν 1: J −1 as detailed in Appendix 7.1.…”

Understanding GPU Programming for Statistical Computation: Studies in Massively Parallel Massive Mixtures

“…A central case in point is Bayesian analysis of mixture models with sample sizes in the millions and hundreds of mixture components; these result in massively expensive computations for Markov chain Monte Carlo (MCMC) analysis and/or Bayesian EM (BEM) for local mode search and optimization, but that can easily be cast in the SIMD/SPMD framework. We focus on multivariate normal mixture models under truncated Dirichlet process (TDP) priors (Ishwaran and James 2001), a variant of a model class very popular in density estimation and classification problems in applied statistics and machine learning (e.g., Escobar and West 1995; MacEachern and Müller 1998a, 1998b; Teh et al 2006; Ji et al 2009). Applied, substantive variants involving heavier-tailed or skewed mixture components, or hierarchical extensions in which component densities are nonnormal, themselves modeled as mixtures of normals, add detail to the computations but do not impact on the main themes and results of interest here.…”

Efficient Classification-Based Relabeling in Mixture Models

“…Escobar and West, 1995; MacEachern and Müller, 1998a,b; Ishwaran and James, 2001; Chan et al, 2008; Ji et al, 2009). As in the previous example, the Gibbs sampler was run for thousands of iterations to ensure convergence; this was followed by many local searches using Bayesian EM to identify local posterior modes and so fix a reference Θ R as the highest posterior mode so identified.…”