Bayesian Analysis

Modelling and Computation Using NCoRM Mixtures for Density Regression

Jim Griffin and Fabrizio Leisen

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Normalized compound random measures are flexible nonparametric priors for related distributions. We consider building general nonparametric regression models using normalized compound random measure mixture models. Posterior inference is made using a novel pseudo-marginal Metropolis-Hastings sampler for normalized compound random measure mixture models. The algorithm makes use of a new general approach to the unbiased estimation of Laplace functionals of compound random measures (which includes completely random measures as a special case). The approach is illustrated on problems of density regression.

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Bayesian Anal., Volume 13, Number 3 (2018), 897-916.

First available in Project Euclid: 26 October 2017

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dependent random measures mixture models multivariate Lévy measures pseudo-marginal samplers Poisson estimator

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Griffin, Jim; Leisen, Fabrizio. Modelling and Computation Using NCoRM Mixtures for Density Regression. Bayesian Anal. 13 (2018), no. 3, 897--916. doi:10.1214/17-BA1072.

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  • Andrieu and Vihola (2016). “Establishing some order amongst exact approximations of MCMCs.” Annals of Applied Probability, 26: 2661–2696.
  • Andrieu, C. and Roberts, G. O. (2009). “The pseudo-marginal approach for efficient Monte Carlo computations.” Annals of Statistics, 37: 697–725.
  • Arbel, J. and Prünster, I. (2016). “A moment-matching Ferguson & Klass algorithm.” Statistics and Computing, 27: 3–17.
  • Atchadé, Y. F. and Rosenthal, J. S. (2005). “On Adaptive Markov Chain Monte Carlo Algorithms.” Bernoulli, 11: 815–828.
  • Brix, A. (1999). “Generalised gamma measures and shot-noise Cox processes.” Advances in Applied Probability, 31: 929–953.
  • Chen, C., Rao, V. A., Buntine, W., and Teh, Y. W. (2013). “Dependent Normalized Random Measures.” In Proceedings of the International Conference on Machine Learning.
  • De Iorio, M., Müller, P., Rosner, G. L., and MacEachern, S. N. (2004). “An ANOVA model for dependent random measures.” Journal of the American Statistical Association, 99: 205–215.
  • Doucet, A., Pitt, M., Deligiannidis, G., and Kohn, R. (2015). “Efficient Implementation of Markov chain Monte Carlo when Using an Unbiased Likelihood Estimator.” Biometrika, 102: 295–313.
  • Dunson, D. B. (2010). “Nonparametric Bayes applications to biostatistics.” In Bayesian Nonparametrics, 223–273. Cambridge University Press.
  • Favaro, S. and Teh, Y. W. (2013). “MCMC for Normalized Random Measure Mixture Models.” Statistical Science, 28: 335–359.
  • Fearnhead, P., Papaspiliopoulos, O., Roberts, G. O., and Stuart, A. (2010). “Random-weight particle filtering of continuous time processes.” Journal of the Royal Statistical Society, Series B, 74: 497–512.
  • Foti, N. and Williamson, S. (2012). “Slice sampling normalized kernel-weighted completely random measure mixture models.” In Advances in Neural Information Processing Systems 25, 2240–2248.
  • Griffin, J. E. (2011). “The Ornstein-Uhlenback Dirichlet process and other time-varying processes for Bayesian nonparametric inference.” Journal of Statistical Planning and Inference, 141: 3648–3664.
  • Griffin, J. E., Kolossiatis, M., and Steel, M. F. J. (2013). “Comparing Distributions By Using Dependent Normalized Random-Measure Mixtures.” Journal of the Royal Statistical Society, Series B, 75: 499–529.
  • Griffin, J. and Leisen, F. (2017). “Appendix of “Modelling and computation using NCoRM mixtures for density regression”.” Bayesian Analysis.
  • Griffin, J. E. and Leisen, F. (2017). “Compound Random Measures and their use in Bayesian nonparametrics.” Journal of the Royal Statistical Society, Series B, 79: 525–545.
  • Griffin, J. E. and Walker, S. G. (2011). “Posterior simulation of normalized random measure mixtures.” Journal of Computational and Graphical Statistics, 20: 241–259.
  • James, L. F., Lijoi, A., and Prünster, I. (2009). “Posterior Analysis for Normalized Random Measures with Independent Increments.” Scandinavian Journal of Statistics, 36: 76–97.
  • Leisen, F. and Lijoi, A. (2011). “Vectors of Poisson-Dirichlet processes.” Journal of Multivariate Analysis, 102: 482–495.
  • Leisen, F., Lijoi, A., and Spano, D. (2013). “A Vector of Dirichlet processes.” Electronic Journal of Statistics, 7: 62–90.
  • Lijoi, A. and Nipoti, B. (2014). “A class of hazard rate mixtures for combining survival data from different experiments.” Journal of the American Statistical Association, 109: 802–814.
  • Lijoi, A., Nipoti, B., and Prünster, I. (2014a). “Bayesian inference with dependent normalized completely random measures.” Bernoulli, 20: 1260–1291.
  • Lijoi, A., Nipoti, B., and Prünster, I. (2014b). “Dependent mixture models: clustering and borrowing information.” Computational Statistics and Data Analysis, 71: 417–433.
  • Lijoi, A. and Prünster, I. (2010). “Models beyond the Dirichlet Process.” In Bayesian Nonparametrics, 80–136. Cambridge University Press.
  • Lyne, A.-M., Girolami, M., Strathmann, H., Simpson, D., and Atchade, Y. (2015). “On Russian roulette estimates for Bayesian inference with doubly-intractable likelihoods.” Statistical Science, 30: 443–467.
  • MacEachern, S. N. (1999). “Dependent nonparametric processes.” In ASA Proceedings of the Section on Bayesian Statistical Science.
  • Müller, P. and Rosner, G. (1997). “A Bayesian population model with hierarchical mixture priors applied to blood count data.” Journal of the American Statistical Association, 92: 1279–1292.
  • Neal, R. M. (2000). “Markov chain sampling methods for Dirichlet process mixture models.” Journal of Computational and Graphical Statistics, 9: 249–265.
  • Papaspiliopoulos, O. (2009). “A methodological framework for Monte Carlo probabilistic inference for diffusion processes.” In Bayesian Time Series Models, 91–112. Cambridge University Press.
  • Ranganath, R. and Blei, D. M. (2017). “Correlated Random Measures.” Journal of the American Statistical Association, to appear.
  • Regazzini, E., Lijoi, A., and Prünster, I. (2003). “Distributional results for means of normalized random measures with independent increments.” Annals of Statistics, 31: 560–585.
  • Rhee, C.-H. and Glynn, P. W. (2015). “Unbiased estimation with square root convergence for SDE models.” Operations Research, 63: 1026–1043.
  • Rodriguez, A. and Dunson, D. B. (2011). “Nonparametric Bayesian models through probit stick-breaking processes.” Bayesian Analysis, 6: 145–178.
  • Sethuraman, J. (1994). “A constructive definition of Dirichlet priors.” Statistica Sinica, 4: 639–650.
  • Silverman, B. W. (1985). “Some aspects of the spline smoothing approach to non-parametric curve fitting.” Journal of the Royal Statistical Society, Series B, 47: 1–52.
  • Todeschini, A. and Caron, F. (2016). “Exchangeable random measures for sparse and modular graphs with overlapping communities.” ArXiv: 1602.0211.
  • Yu, Y. and Meng, X.-L. (2011). “To Center or Not to Center: That is Not the Question – An Ancillarity-Sufficiency Interweaving Strategy (ASIS) for Boosting MCMC Efficiency.” Journal of Computational and Graphical Statistics, 20: 531–570.
  • Zhu, W. and Leisen, F. (2015). “A multivariate extension of a vector of Poisson-Dirichlet processes.” Journal of Nonparametric Statistics, 27: 89–105.

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