Bayesian Inference for Non-Markovian Point Processes

The Bayesian approach to statistical inference has in recent years become very popular, especially in the analysis of complex data sets. This is largely due to the development of Markov chain Monte Carlo methods, which expand the scope of application of Bayesian methods considerably. In this paper, we review the Bayesian contributions to inference for point processes. We focus on non-Markovian processes, specifically Poisson and related models, doubly stochastic models, and cluster models. We also discuss Bayesian model selection for these models and give examples in which Bayes factors are applied both directly and indirectly through a reversible jump algorithm.

[1]  Eva Bjørn Vedel Jensen,et al.  Bayesian analysis of spatial point processes in the neighbourhood of Voronoi networks , 2007, Stat. Comput..

[2]  E. B. Jensen,et al.  Asymptotic Palm likelihood theory for stationary point processes , 2013 .

[3]  Stephen M. Smith,et al.  Investigations into resting-state connectivity using independent component analysis , 2005, Philosophical Transactions of the Royal Society B: Biological Sciences.

[4]  Albert Y. Lo,et al.  Bayesian nonparametric statistical inference for Poisson point processes , 1982 .

[5]  Rasmus Waagepetersen,et al.  Convergence of posteriors for discretized log Gaussian Cox processes , 2004 .

[6]  Ian W. McKeague,et al.  Perfect Sampling for Point Process Cluster Modelling , 2002 .

[7]  David R. Cox,et al.  The statistical analysis of series of events , 1966 .

[8]  D. Gamerman.  A dynamic approach to the statistical analysis of point processes , 1992 .

[9]  R. Wolpert,et al.  Likelihood-based inference for Matérn type-III repulsive point processes , 2009, Advances in Applied Probability.

[10]  Vicki M. Bier,et al.  A natural conjugate prior for the nonhomogeneous poisson process with an exponential intensity function , 1999 .

[11]  J. Besag,et al.  Statistical Analysis of Spatial Point Patterns by Means of Distance Methods , 1976 .

[12]  Ilkka Taskinen.  Cluster priors in the Bayesian modelling of fMRI data , 2001 .

[13]  Adrian Baddeley,et al.  Markov properties of cluster processes , 1996, Advances in Applied Probability.

[14]  J. Møller,et al.  A CASE STUDY ON POINT PROCESS MODELLING IN DISEASE MAPPING , 2011 .

[15]  J. Møller,et al.  Log Gaussian Cox Processes , 1998 .

[16]  R. Waagepetersen,et al.  Modern Statistics for Spatial Point Processes * , 2007 .

[17]  R Core Team,et al.  R: A language and environment for statistical computing. , 2014 .

[18]  M. Bartlett.  The Spectral Analysis of Point Processes , 1963 .

[19]  Y. Ogata.  Seismicity Analysis through Point-process Modeling: A Review , 1999 .

[20]  P. Guttorp.  Analysis of event-based precipitation data with a view toward modeling , 1988 .

[21]  Lynn Kuo,et al.  Bayesian Computation for Nonhomogeneous Poisson Processes in Software Reliability , 1996 .

[22]  O. Aalen.  Nonparametric Inference for a Family of Counting Processes , 1978 .

[23]  Simon Godsill,et al.  Poisson point process modeling for polyphonic music transcription. , 2007, The Journal of the Acoustical Society of America.

[24]  H. Rue,et al.  Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations , 2009 .

[25]  Scott A. Sisson,et al.  Statistical Inference and Simulation for Spatial Point Processes , 2005 .

[26]  P. Diggle.  A Kernel Method for Smoothing Point Process Data , 1985 .

[27]  T. Thorarinsdottir,et al.  A Spatio‐Temporal Model for Functional Magnetic Resonance Imaging Data – with a View to Resting State Networks , 2007 .

[28]  Yongdai Kim.  NONPARAMETRIC BAYESIAN ESTIMATORS FOR COUNTING PROCESSES , 1999 .

[29]  D. Cox.  Some Statistical Methods Connected with Series of Events , 1955 .

[30]  Dietrich Stoyan,et al.  Parameter Estimation and Model Selection for Neyman‐Scott Point Processes , 2008, Biometrical journal. Biometrische Zeitschrift.

[31]  A. Raftery,et al.  Bayes Factors for Non‐Homogeneous Poisson Processes with Vague Prior Information , 1986 .

[32]  R. Wolpert,et al.  LIKELIHOOD-BASED INFERENCE FOR MAT´ ERN TYPE III RE- PULSIVE POINT PROCESSES , 2009 .

[33]  D. Cox,et al.  The statistical analysis of series of events , 1966 .

[34]  Andrew B. Lawson,et al.  Spatial cluster modelling , 2002 .

[35]  H. Jeffreys.  Some Tests of Significance, Treated by the Theory of Probability , 1935, Mathematical Proceedings of the Cambridge Philosophical Society.

[36]  Lucien Le Cam,et al.  A Stochastic Description of Precipitation , 1961 .

[37]  Daryl J. Daley,et al.  An Introduction to the Theory of Point Processes , 2013 .

[38]  N. Hartvig,et al.  A Stochastic Geometry Model for Functional Magnetic Resonance Images , 2002 .

[39]  Agus Salim,et al.  Extensions of the Bartlett-Lewis model for rainfall processes , 2003 .

[40]  Adrian Baddeley,et al.  Centrum Voor Wiskunde En Informatica Probability, Networks and Algorithms Probability, Networks and Algorithms Extrapolating and Interpolating Spatial Patterns Extrapolating and Interpolating Spatial Patterns , 2022 .

[41]  J. Heikkinen,et al.  Non‐parametric Bayesian Estimation of a Spatial Poisson Intensity , 1998 .

[42]  R. Waagepetersen.  An Estimating Function Approach to Inference for Inhomogeneous Neyman–Scott Processes , 2007, Biometrics.

[43]  DiscussionThe MAP3S/RAINE precipitation chemistry network: Statistical overview for the period 1976–1980 and wet deposition variability as observed by MAP3S☆☆☆ , 1983 .

[44]  Tore Schweder,et al.  Likelihood-based inference for clustered line transect data , 2006 .

[45]  A. Kottas.  Dirichlet Process Mixtures of Beta Distributions , with Applications to Density and Intensity Estimation , 2006 .

[46]  P. Green,et al.  On Bayesian Analysis of Mixtures with an Unknown Number of Components (with discussion) , 1997 .

[47]  R. Wolpert,et al.  Perfect simulation and moment properties for the Matérn type III process , 2010 .

[48]  Peter Guttorp,et al.  Stochastic modeling of rainfall , 1996 .

[49]  Adrian E. Raftery,et al.  Classification of Mixtures of Spatial Point Processes via Partial Bayes Factors , 2005 .

[50]  N. Cressie,et al.  Asymptotic Properties of Estimators for the Parameters of Spatial Inhomogeneous Poisson Point Processes , 1994, Advances in Applied Probability.

[51]  P. Hobbs,et al.  Rainbands, Precipitation Cores and Generating Cells in a Cyclonic Storm , 1978 .

[52]  R. Wolpert,et al.  Poisson/gamma random field models for spatial statistics , 1998 .

[53]  A. Kottas,et al.  Bayesian mixture modeling for spatial Poisson process intensities, with applications to extreme value analysis , 2007 .

[54]  D. Stoyan,et al.  Statistical Analysis and Modelling of Spatial Point Patterns , 2008 .

[55]  G. Shedler,et al.  Simulation of Nonhomogeneous Poisson Processes by Thinning , 1979 .

[56]  Albert Y. Lo,et al.  On a class of Bayesian nonparametric estimates: II. Hazard rate estimates , 1989 .

[57]  A. Gelfand,et al.  Handbook of spatial statistics , 2010 .

[58]  Ryan P. Adams,et al.  Tractable nonparametric Bayesian inference in Poisson processes with Gaussian process intensities , 2009, ICML '09.

[59]  P. Green.  Reversible jump Markov chain Monte Carlo computation and Bayesian model determination , 1995 .

[60]  J. Delleur,et al.  A stochastic cluster model of daily rainfall sequences , 1981 .