## Bernoulli

• Bernoulli
• Volume 19, Number 5B (2013), 2590-2626.

### A conjugate class of random probability measures based on tilting and with its posterior analysis

John W. Lau

#### Abstract

This article constructs a class of random probability measures based on exponentially and polynomially tilting operated on the laws of completely random measures. The class is proved to be conjugate in that it covers both prior and posterior random probability measures in the Bayesian sense. Moreover, the class includes some common and widely used random probability measures, the normalized completely random measures (James (Poisson process partition calculus with applications to exchangeable models and Bayesian nonparametrics (2002) Preprint), Regazzini, Lijoi and Prünster (Ann. Statist. 31 (2003) 560–585), Lijoi, Mena and Prünster (J. Amer. Statist. Assoc. 100 (2005) 1278–1291)) and the Poisson–Dirichlet process (Pitman and Yor (Ann. Probab. 25 (1997) 855–900), Ishwaran and James (J. Amer. Statist. Assoc. 96 (2001) 161–173), Pitman (In Science and Statistics: A Festschrift for Terry Speed (2003) 1–34 IMS)), in a single construction. We describe an augmented version of the Blackwell–MacQueen Pólya urn sampling scheme (Blackwell and MacQueen (Ann. Statist. 1 (1973) 353–355)) that simplifies implementation and provide a simulation study for approximating the probabilities of partition sizes.

#### Article information

Source
Bernoulli, Volume 19, Number 5B (2013), 2590-2626.

Dates
First available in Project Euclid: 3 December 2013

https://projecteuclid.org/euclid.bj/1386078614

Digital Object Identifier
doi:10.3150/12-BEJ467

Mathematical Reviews number (MathSciNet)
MR3160565

Zentralblatt MATH identifier
06254573

#### Citation

Lau, John W. A conjugate class of random probability measures based on tilting and with its posterior analysis. Bernoulli 19 (2013), no. 5B, 2590--2626. doi:10.3150/12-BEJ467. https://projecteuclid.org/euclid.bj/1386078614

#### References

• [1] Aldous, D.J. (1985). Exchangeability and related topics. In École d’Été de Probabilités de Saint-Flour, XIII—1983. Lecture Notes in Math. 1117 1–198. Berlin: Springer.
• [2] Arratia, R., Barbour, A.D. and Tavaré, S. (2003). Logarithmic Combinatorial Structures: A Probabilistic Approach. EMS Monographs in Mathematics. Zürich: European Mathematical Society (EMS).
• [3] Blackwell, D. and MacQueen, J.B. (1973). Ferguson distributions via Pólya urn schemes. Ann. Statist. 1 353–355.
• [4] Brix, A. (1999). Generalized gamma measures and shot-noise Cox processes. Adv. in Appl. Probab. 31 929–953.
• [5] Cifarelli, D.M. and Regazzini, E. (1979). Considerazioni generali sull’impostazione bayesiana di problemi non parametrici. Le medie associative nel contesto del processo aleatorio di Dirichlet. Parte I. Riv. Mat. Sci. Econom. Social. 2 39–52.
• [6] Cifarelli, D.M. and Regazzini, E. (1979). Considerazioni generali sull’impostazione bayesiana di problemi non parametrici. Le medie associative nel contesto del processo aleatorio di Dirichlet. Parte II. Riv. Mat. Sci. Econom. Social. 2 95–111.
• [7] Cifarelli, D.M. and Regazzini, E. (1990). Distribution functions of means of a Dirichlet process. Ann. Statist. 18 429–442. [Correction Ann. Statist. 22 (1994) 1633–1634].
• [8] Daley, D.J. and Vere-Jones, D. (2008). An Introduction to the Theory of Point Processes. Vol. II: General Theory and Structure, 2nd ed. Probability and Its Applications (New York). New York: Springer.
• [9] Devroye, L. (1986). Nonuniform Random Variate Generation. New York: Springer.
• [10] Doksum, K. (1974). Tailfree and neutral random probabilities and their posterior distributions. Ann. Probab. 2 183–201.
• [11] Dykstra, R.L. and Laud, P. (1981). A Bayesian nonparametric approach to reliability. Ann. Statist. 9 356–367.
• [12] Ferguson, T.S. (1973). A Bayesian analysis of some nonparametric problems. Ann. Statist. 1 209–230.
• [13] Griffin, J.E. and Steel, M.F.J. (2006). Order-based dependent Dirichlet processes. J. Amer. Statist. Assoc. 101 179–194.
• [14] Griffiths, R.C. and Lessard, S. (2005). Ewens’ sampling formula and related formulae: Combinatorial proofs, extensions to variable population size and applications to ages of alleles. Theor. Popul. Biol. 68 167–177.
• [15] Hougaard, P. (1986). Survival models for heterogeneous populations derived from stable distributions. Biometrika 73 387–396.
• [16] Ishwaran, H. and James, L.F. (2001). Gibbs sampling methods for stick-breaking priors. J. Amer. Statist. Assoc. 96 161–173.
• [17] Ishwaran, H. and James, L.F. (2003). Generalized weighted Chinese restaurant processes for species sampling mixture models. Statist. Sinica 13 1211–1235.
• [18] James, L.F. (2002). Poisson process partition calculus with applications to exchangeable models and Bayesian nonparametrics. Preprint. Available at http://arxiv.org/abs/math/0205093.
• [19] James, L.F. (2005). Bayesian Poisson process partition calculus with an application to Bayesian Lévy moving averages. Ann. Statist. 33 1771–1799.
• [20] James, L.F. (2005). Functionals of Dirichlet processes, the Cifarelli–Regazzini identity and beta-gamma processes. Ann. Statist. 33 647–660.
• [21] James, L.F., Lijoi, A. and Prünster, I. (2006). Conjugacy as a distinctive feature of the Dirichlet process. Scand. J. Statist. 33 105–120.
• [22] James, L.F., Lijoi, A. and Prünster, I. (2009). Posterior analysis for normalized random measures with independent increments. Scand. J. Stat. 36 76–97.
• [23] Kallenberg, O. (1983). Random Measures, 3rd ed. Berlin: Akademie-Verlag.
• [24] Kallenberg, O. (2001). Foundations of Modern Probability, 2nd ed. New York: Springer.
• [25] Kingman, J.F.C. (1967). Completely random measures. Pacific J. Math. 21 59–78.
• [26] Kingman, J.F.C. (1993). Poisson Processes. Oxford Studies in Probability 3. New York: Oxford Univ. Press.
• [27] Kingman, J.F.C., Taylor, S.J., Hawkes, A.G., Walker, A.M., Cox, D.R., Smith, A.F.M., Hill, B.M., Burville, P.J. and Leonard, T. (1975). Random discrete distribution. J. Roy. Statist. Soc. Ser. B 37 1–22. With a discussion by S. J. Taylor, A. G. Hawkes, A. M. Walker, D. R. Cox, A. F. M. Smith, B. M. Hill, P. J. Burville, T. Leonard and a reply by the author.
• [28] Lau, J.W. and Cripps, E. (2012). Bayesian non-parametric mixtures of $\operatorname{GARCH}(1,1)$ models. J. Probab. Stat. 2012 Art. ID 167431.
• [29] Lau, J.W. and So, M.K.P. (2008). Bayesian mixture of autoregressive models. Comput. Statist. Data Anal. 53 38–60.
• [30] Lau, J.W. and So, M.K.P. (2011). A Monte Carlo Markov chain algorithm for a class of mixture time series models. Stat. Comput. 21 69–81.
• [31] Lijoi, A., Mena, R.H. and Prünster, I. (2005). Bayesian nonparametric analysis for a generalized Dirichlet process prior. Stat. Inference Stoch. Process. 8 283–309.
• [32] Lijoi, A., Mena, R.H. and Prünster, I. (2005). Hierarchical mixture modeling with normalized inverse-Gaussian priors. J. Amer. Statist. Assoc. 100 1278–1291.
• [33] Lijoi, A., Mena, R.H. and Prünster, I. (2007). Bayesian nonparametric estimation of the probability of discovering new species. Biometrika 94 769–786.
• [34] Lijoi, A., Mena, R.H. and Prünster, I. (2007). Controlling the reinforcement in Bayesian non-parametric mixture models. J. R. Stat. Soc. Ser. B Stat. Methodol. 69 715–740.
• [35] Lijoi, A. and Prünster, I. (2010). Models beyond the Dirichlet process. In Bayesian Nonparametrics (N.L. Hjort, C. Holmes, P. Müller and S.G. Walker, eds.). Camb. Ser. Stat. Probab. Math. 80–136. Cambridge: Cambridge Univ. Press.
• [36] Lo, A.Y. (1984). On a class of Bayesian nonparametric estimates. I. Density estimates. Ann. Statist. 12 351–357.
• [37] Lo, A.Y. (1991). A characterization of the Dirichlet process. Statist. Probab. Lett. 12 185–187.
• [38] Lo, A.Y., Brunner, L.J. and Chan, A.T. (1996). Weighted Chinese restaurant processes and Bayesian mixture models. Research report, Hong Kong Univ. Science and Technology. Available at http://www.utstat.utoronto.ca/~brunner/papers/wcr96.pdf.
• [39] Lo, A.Y. and Weng, C.S. (1989). On a class of Bayesian nonparametric estimates. II. Hazard rate estimates. Ann. Inst. Statist. Math. 41 227–245.
• [40] Pitman, J. (1995). Exchangeable and partially exchangeable random partitions. Probab. Theory Related Fields 102 145–158.
• [41] Pitman, J. (2003). Poisson–Kingman partitions. In Statistics and Science: A Festschrift for Terry Speed. Institute of Mathematical Statistics Lecture Notes—Monograph Series 40 1–34. Beachwood, OH: IMS.
• [42] Pitman, J. (2006). Combinatorial Stochastic Processes. Lecture Notes in Math. 1875. Berlin: Springer. Lectures from the 32nd Summer School on Probability Theory held in Saint-Flour, July 7–24, 2002, With a foreword by Jean Picard.
• [43] Pitman, J. and Yor, M. (1997). The two-parameter Poisson–Dirichlet distribution derived from a stable subordinator. Ann. Probab. 25 855–900.
• [44] Regazzini, E., Lijoi, A. and Prünster, I. (2003). Distributional results for means of normalized random measures with independent increments. Ann. Statist. 31 560–585. Dedicated to the memory of Herbert E. Robbins.
• [45] Wolpert, R.L. and Ickstadt, K. (1998). Poisson/gamma random field models for spatial statistics. Biometrika 85 251–267.