The Annals of Statistics

Bayesian Poisson process partition calculus with an application to Bayesian Lévy moving averages

Lancelot F. James

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Abstract

This article develops, and describes how to use, results concerning disintegrations of Poisson random measures. These results are fashioned as simple tools that can be tailor-made to address inferential questions arising in a wide range of Bayesian nonparametric and spatial statistical models. The Poisson disintegration method is based on the formal statement of two results concerning a Laplace functional change of measure and a Poisson Palm/Fubini calculus in terms of random partitions of the integers {1,…,n}. The techniques are analogous to, but much more general than, techniques for the Dirichlet process and weighted gamma process developed in [Ann. Statist. 12 (1984) 351–357] and [Ann. Inst. Statist. Math. 41 (1989) 227–245]. In order to illustrate the flexibility of the approach, large classes of random probability measures and random hazards or intensities which can be expressed as functionals of Poisson random measures are described. We describe a unified posterior analysis of classes of discrete random probability which identifies and exploits features common to all these models. The analysis circumvents many of the difficult issues involved in Bayesian nonparametric calculus, including a combinatorial component. This allows one to focus on the unique features of each process which are characterized via real valued functions h. The applicability of the technique is further illustrated by obtaining explicit posterior expressions for Lévy–Cox moving average processes within the general setting of multiplicative intensity models. In addition, novel computational procedures, similar to efficient procedures developed for the Dirichlet process, are briefly discussed for these models.

Article information

Source
Ann. Statist. Volume 33, Number 4 (2005), 1771-1799.

Dates
First available in Project Euclid: 5 August 2005

Permanent link to this document
http://projecteuclid.org/euclid.aos/1123250229

Digital Object Identifier
doi:10.1214/009053605000000336

Mathematical Reviews number (MathSciNet)
MR2166562

Subjects
Primary: 62G05: Estimation
Secondary: 62F15: Bayesian inference

Keywords
Cumulants inhomogeneous Poisson process Lévy measure multiplicative intensity model generalized gamma process weighted Chinese restaurant

Citation

James, Lancelot F. Bayesian Poisson process partition calculus with an application to Bayesian Lévy moving averages. The Annals of Statistics 33 (2005), no. 4, 1771--1799. doi:10.1214/009053605000000336. http://projecteuclid.org/euclid.aos/1123250229.


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References

  • Aalen, O. O. (1975). Statistical inference for a family of counting processes. Ann. Statist. 6 701--726.
  • Andersen, P. K., Borgan, Ø., Gill, R. D. and Keiding, N. (1993). Statistical Models Based on Counting Processes. Springer, New York.
  • Antoniak, C. E. (1974). Mixtures of Dirichlet processes with applications to Bayesian nonparametric problems. Ann. Statist. 2 1152--1174.
  • Barndorff-Nielsen, O. E. and Shephard, N. (2001). Non-Gaussian Ornstein--Uhlenbeck-based models and some of their uses in financial economics. J. R. Stat. Soc. Ser. B Stat. Methodol. 63 167--241.
  • Blackwell, D. and MacQueen, J. B. (1973). Ferguson distributions via Pólya urn schemes. Ann. Statist. 1 353--355.
  • Brix, A. (1999). Generalized Gamma measures and shot-noise Cox processes. Adv. in Appl. Probab. 31 929--953.
  • Daley, D. J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York.
  • Doksum, K. A. (1974). Tailfree and neutral random probabilities and their posterior distributions. Ann. Probab. 2 183--201.
  • Draghici, L. and Ramamoorthi, R. V. (2003). Consistency of Dykstra--Laud priors. Sankhyā 65 464--481.
  • Dykstra, R. L. and Laud, P. W. (1981). A Bayesian nonparametric approach to reliability. Ann. Statist. 9 356--367.
  • Escobar, M. D. (1994). Estimating normal means with a Dirichlet process prior. J. Amer. Statist. Assoc. 89 268--277.
  • Ewens, W. J. (1972). The sampling theory of selectively neutral alleles. Theoret. Population Biology 3 87--112.
  • Ferguson, T. S. (1973). A Bayesian analysis of some nonparametric problems Ann. Statist. 1 209--230.
  • Hjort, N. L. (1990). Nonparametric Bayes estimators based on beta processes in models for life history data. Ann. Statist. 18 1259--1294.
  • Ishwaran, H. and James, L. F. (2003). Generalized weighted Chinese restaurant processes for species sampling mixture models. Statist. Sinica 13 1211--1235.
  • Ishwaran, H. and James, L. F. (2004). Computational methods for multiplicative intensity models using weighted gamma processes: Proportional hazards, marked point processes and panel count data. J. Amer. Statist. Assoc. 99 175--190.
  • Jacod, J. (1975). Multivariate point process: Predictable projection, Radon--Nikodym derivatives, representation of martingales. Z. Wahrsch. Verw. Gebiete 31 235--253.
  • James, L. F. (2002). Poisson process partition calculus with applications to exchangeable models and Bayesian nonparametrics. arXiv.org/math.PR/0205093.
  • James, L. F. (2003). Bayesian calculus for Gamma processes with applications to semiparametric intensity models. Sankhyā 65 196--223.
  • James, L. F. (2005). Poisson calculus for spatial neutral to the right processes. Ann. Statist. To appear.
  • Kallenberg, O. (1986). Random Measures, 4th ed. Akademie-Verlag and Academic Press, Berlin and London.
  • Kallenberg, O. (2002). Foundations of Modern Probability. Probability and Its Applications, 2nd ed. Springer, New York.
  • Kingman, J. F. C. (1967). Completely random measures. Pacific J. Math. 21 59--78.
  • Kingman, J. F. C. (1975). Random discrete distributions (with discussion). J. Roy. Statist. Soc. Ser. B 37 1--22.
  • Kingman, J. F. C. (1978). Uses of exchangeability. Ann. Probab. 6 183--197.
  • Kingman, J. F. C. (1993). Poisson Processes. Oxford Univ. Press.
  • Küchler, U. and Sørensen, M. (1997). Exponential Families of Stochastic Processes. Springer, New York.
  • Liu, J. S. (1996). Nonparametric hierarchichal Bayes via sequential imputations. Ann. Statist. 24 911--930.
  • Lo, A. Y. (1982). Bayesian nonparametric statistical inference for Poisson point processes. Z. Wahrsch. Verw. Gebiete 59 55--66.
  • Lo, A. Y. (1984). On a class of Bayesian nonparametric estimates. I. Density estimates. Ann. Statist. 12 351--357.
  • Lo, A. Y., Brunner, L. J. and Chan, A. T. (1996). Weighted Chinese restaurant processes and Bayesian mixture model. Research report, Hong Kong Univ. Science and Technology.
  • Lo, A. Y. and Weng, C.-S. (1989). On a class of Bayesian nonparametric estimates. II. Hazard rates estimates. Ann. Inst. Statist. Math. 41 227--245.
  • Nieto-Barajas, L. E., Prünster, I. and Walker, S. G. (2004). Normalized random measures driven by increasing additive processes. Ann. Statist. 32 2343--2360.
  • Nieto-Barajas, L. E. and Walker, S. G. (2004). Bayesian nonparametric survival analysis via Lévy driven Markov processes. Statist. Sinica 14 1127--1146.
  • Perman, M., Pitman, J. and Yor, M. (1992). Size-biased sampling of Poisson point processes and excursions. Probab. Theory Related Fields 92 21--39.
  • Pitman, J. (1995). Exchangeable and partially exchangeable random partitions. Probab. Theory Related Fields 102 145--158.
  • Pitman, J. (1996). Some developments of the Blackwell--MacQueen urn scheme. In Statistics, Probability and Game Theory (T. S. Ferguson, L. S. Shapley and J. B. MacQueen, eds.) 245--267. IMS, Hayward, CA.
  • Pitman, J. (2002). Combinatorial stochastic processes. Technical Report 621, Dept. Statistics, Univ. California, Berkeley. Lecture notes for Saint-Flour course, July 2002. Available at www.stat.berkeley.edu/~pitman.
  • Pitman, J. (2003). Poisson--Kingman partitions. In Science and Statistics: A Festschrift for Terry Speed (D. R. Goldstein, ed.) 1--34. IMS, Beachwood, OH.
  • Regazzini, E., Lijoi, A. and Prünster, I. (2003). Distributional results for means of normalized random measures with independent increments. Ann. Statist. 31 560--585.
  • Wolpert, R. L. and Ickstadt, K. (1988). Poisson/Gamma random field models for spatial statistics. Biometrika 85 251--267.
  • Wolpert, R. L. and Ickstadt, K. (1998). Simulation of Levy random fields. Practical Nonparametric and Semiparametric Bayesian Statistics. Lecture Notes in Statist. 133 227--242. Springer, New York.