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 immunofluorescence histology data demonstrate the models and computational methodology.

Keywords: Bayesian Computation, Blocked Gibbs Sampler, Dirichlet Process Mixture Model, Spatial Inhomogeneous Poisson Process, Unobserved Point Process.

Supporting material is available, in terms of additional information on examples, and software:

• Movies that illustrate posterior uncertainties about intensity surfaces in the day-1 example from the paper:
  • contour plots of successive samples of the DP mixture PDF across a series of MCMC iterations.
  • image plots of the above.

  • contour plots of MCMC-based estimate of predictive PDF as it is updated through a series of MCMC iterations.
  • image plots of the above.

  • Plots of pixel probabilities representing presence/absence of cell-based fluorescence as they vary over a series of MCMC iterations.
  • Sampled spatial point pattern -- i.e., locations of cells -- corresponding to the above probabilities.

• Matlab code and data for examples from the paper, as well as supporting Matlab code for general use in other model contexts. This is freely available to interested researchers. The Matlab script file spatmix_day1_master.m there contains commands to read in data and execute the analysis, based on one of the example jpg images from the paper. The full directory of code and data is also available for download in compressed format.

  This software is made freely available to any interested user. The authors can provide no support nor assistance with implementations beyond the details and examples here, nor extensions of the code for other purposes. The download has been tested to confirm all details are operational as described here. It is understood by the user that neither the authors nor Duke University bear any responsibility nor assume any liability for any end-use of this software. It is expected that appropriate credit/acknowledgement be given should the software be included as an element in other software development or in publications.