Bayesian Analysis

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

Chunlin Ji, Thomas B. Kepler, Daniel Merl, and Mike West

Full-text: Open access


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.

Article information

Bayesian Anal., Volume 4, Number 2 (2009), 297-315.

First available in Project Euclid: 22 June 2012

Permanent link to this document

Digital Object Identifier
doi: 10.1214/09-BA411

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Bayesian computation blocked Gibbs sampler Dirichlet process mixture model inhomogeneous Poisson process unobserved point process


Ji, Chunlin; Merl, Daniel; Kepler, Thomas B.; West, Mike. Spatial mixture modelling for unobserved point processes: examples in immunofluorescence histology. Bayesian Anal. 4 (2009), no. 2, 297--315. doi:10.1214/09-BA411.

Export citation


  • Daley, D. and Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes (2nd edn.). New York: Springer Verlag.
  • Diggle, P. (2003). Statistical Analysis of Spatial Point Patterns. London: Arnold.
  • Escobar, M. and West, M. (1995). "Bayesian density estimation and inference using mixtures." Journal of the American Statistical Association, 90: 577–588.
  • –- (1998). "Computing nonparametric hierarchical models." In Dey, D., Mueller, P., and Sinha, D. (eds.), Practical Nonparametric and Semiparametric Bayesian Statistics, 1–22. New York: Springer Verlag.
  • Ishwaran, H. and James, L. (2001). "Gibbs Sampling Methods for Stick-Breaking Priors." Journal of the American Statistical Association, 96: 161–173.
  • Kottas, A. and Sanso, B. (2007). "Bayesian Mixture Modeling for Spatial Poisson Process Intensities, with Applications to Extreme Value Analysis." Journal of Statistical Planning and Inference (Special Issue on Bayesian Inference for Stochastic Processes), 137: 3151–3163.
  • MacEachern, S. (1994). "Estimating normal means with a conjugate style Dirichlet process prior." Communications in Statistics: Simulation and Computation, 23: 727–741.
  • MacEachern, S. and Mueller, P. (1998). "Estimating mixture of Dirichlet process models." Journal of Computational and Graphical Statististics, 7: 223–238.
  • MacEachern, S. N. (1998). "Computational methods for mixture of Dirichlet process models." In Dey, D., Mueller, P., and Sinha, D. (eds.), Practical Nonparametric and Semiparametric Bayesian Statistics, 23–43. New York: Springer Verlag.
  • Moller, J. and Waagepetersen, R. (2004). Statistical Inference and Simulation for Spatial Point Processes. London: Chapman and Hall.
  • Mueller, P. and Quintana, F. (2004). "Nonparametric Bayesian data analysis." Statistical Science, 19: 95–110.
  • Sethuraman, J. (1994). "A constructive definition of Dirichlet priors." Statistica Sinica, 4: 639–650.
  • West, M., Mueller, P., and Escobar, M. (1994). "Hierarchical priors and mixture models, with application in regression and density estimation." In Freeman, P. and Smith, A. (eds.), Aspects of Uncertainty: A Tribute to D.V. Lindley, 363–386. London: Wiley.
  • Wolpert, R. and Ickstadt, K. (1998). "Poisson/Gamma random field models for spatial statistics." Biometrika, 85: 251–267.