The Annals of Applied Statistics

Using informative priors in the estimation of mixtures over time with application to aerosol particle size distributions

Darren Wraith, Kerrie Mengersen, Clair Alston, Judith Rousseau, and Tareq Hussein

Full-text: Open access


The issue of using informative priors for estimation of mixtures at multiple time points is examined. Several different informative priors and an independent prior are compared using samples of actual and simulated aerosol particle size distribution (PSD) data. Measurements of aerosol PSDs refer to the concentration of aerosol particles in terms of their size, which is typically multimodal in nature and collected at frequent time intervals. The use of informative priors is found to better identify component parameters at each time point and more clearly establish patterns in the parameters over time. Some caveats to this finding are discussed.

Article information

Ann. Appl. Stat. Volume 8, Number 1 (2014), 232-258.

First available in Project Euclid: 8 April 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Bayesian statistics mixture models time series aerosol particle size distribution


Wraith, Darren; Mengersen, Kerrie; Alston, Clair; Rousseau, Judith; Hussein, Tareq. Using informative priors in the estimation of mixtures over time with application to aerosol particle size distributions. Ann. Appl. Stat. 8 (2014), no. 1, 232--258. doi:10.1214/13-AOAS678.

Export citation


  • Aalto, P., Hämeri, K., Becker, E., Weber, R., Salm, J., Makela, J. M., Hoell, C., O’Dowd, C., Karlsson, H., Hansson, H. C., Vakeva, M., Koponen, I. K., Buzoris, G. and Kulmala, M. (2001). Physical characterization of aerosol particles during nucleation events. Tellus 53 344–358.
  • Alston, C. L. and Mengersen, K. L. (2010). Allowing for the effect of data binning in a Bayesian normal mixture model. Comput. Statist. Data Anal. 54 916–923.
  • Alston, C. L., Mengersen, K. L., Robert, C. P., Thompson, J. M., Littlefield, P. J., Perry, D. and Ball, A. J. (2007). Bayesian mixture models in a longitudinal setting for analysing sheep CAT scan images. Comput. Statist. Data Anal. 51 4282–4296.
  • Brooks, S. P. and Gelman, A. (1998). General methods for monitoring convergence of iterative simulations. J. Comput. Graph. Statist. 7 434–455.
  • Caron, F., Davy, M. and Doucet, A. (2007). Generalized Polya urn for time-varying Dirichlet process mixtures. In 23rd Conference on Uncertainty in Artificial Intelligence (UAI’2007). Vancouver, Canada.
  • Celeux, G., Hurn, M. and Robert, C. P. (2000). Computational and inferential difficulties with mixture posterior distributions. J. Amer. Statist. Assoc. 95 957–970.
  • Dal Maso, M., Kulmala, M., Riipinen, I., Wagner, R., Hussein, T., Aalto, P. P. and Lehtinen, K. E. J. (2005). Formation and growth of fresh atmospheric aerosols: Eight years of aerosol size distribution data from SMEAR II, Hyytiala, Finland. Boreal Env. Res. 10 323–336.
  • Dunson, D. B. (2006). Bayesian dynamic modelling of latent trait distributions. Biostatistics 7 551–568.
  • Fahrmeir, L., Kneib, T. and Lang, S. (2004). Penalized structured additive regression for space-time data: A Bayesian perspective. Statist. Sinica 14 731–761.
  • Fernández, C. and Green, P. J. (2002). Modelling spatially correlated data via mixtures: A Bayesian approach. J. R. Stat. Soc. Ser. B Stat. Methodol. 64 805–826.
  • Frühwirth-Schnatter, S. (2001). Markov chain Monte Carlo estimation of classical and dynamic switching and mixture models. J. Amer. Statist. Assoc. 96 194–209.
  • Frühwirth-Schnatter, S. (2006). Finite Mixture and Markov Switching Models. Springer, New York.
  • Gelman, A. (2006). Prior distributions for variance parameters in hierarchical models. Bayesian Anal. 1 515–533 (electronic).
  • Godsill, S., Doucet, A. and West, M. (2001). Maximum a posteriori sequence estimation using Monte Carlo particle filters. Ann. Inst. Statist. Math. 53 82–96.
  • Green, P. J. and Richardson, S. (2002). Hidden Markov models and disease mapping. J. Amer. Statist. Assoc. 97 1055–1070.
  • Gustafson, P. and Walker, L. J. (2003). An extension of the Dirichlet prior for the analysis of longitudinal multinomial data. J. Appl. Stat. 30 293–310.
  • Hoff, P. D. (2003). Nonparametric modelling of hierarchically exchangeable data. Technical report, Dept. Statistics, Univ. Washington.
  • Hussein, T., Hämeri, K., Aalto, P., Paatero, P. and Kulmala, M. (2005). Modal structure and spatial–temporal variations of urban and suburban aerosols in Helsinki–Finland. Atmos. Environ. 39 1655–1668.
  • Ibrahim, J. G., Chen, M.-H. and Sinha, D. (2003). On optimality properties of the power prior. J. Amer. Statist. Assoc. 98 204–213.
  • Jasra, A., Holmes, C. C. and Stephens, D. A. (2005). Markov chain Monte Carlo methods and the label switching problem in Bayesian mixture modeling. Statist. Sci. 20 50–67.
  • Ji, C. (2009). Advances in Bayesian modelling and computation: Spatio-temporal processes, model assessment and advances in Bayesian modelling and computation: Spatio-temporal processes, model assessment and adaptive MCMC. Ph.D. thesis, Dept. Statistical Science, Univ. Duke.
  • Kulmala, M., Vehkamäkia, H., Petäjäa, T., Dal Maso, M., Lauria, A., Birmili, W. and McMurry, P. H. (2004). Formation and growth rates of ultrafine atmospheric particles: A review of observations. J. Aerosol Sci. 35 143–176.
  • MacEachern, S. N. (1999). Dependent nonparametric processes. In ASA Proceedings of the Section on Bayesian Statistical Science. Amer. Statist. Assoc., Alexandria, VA.
  • Marin, J. M., Mengersen, K. and Robert, C. P. (2005). Bayesian modelling and inference on mixtures of distributions. In Handbook of Statistics 25 (D. Dey and C. R. Rao, eds.). Elsevier, Amsterdam.
  • McMurry, P. H. (2000). A review of atmospheric aerosol measurements. Atmos. Environ. 34 1959–1999.
  • Richardson, S. and Green, P. J. (1997). On Bayesian analysis of mixtures with an unknown number of components. J. R. Stat. Soc. Ser. B Stat. Methodol. 59 731–792.
  • Robert, C. P. and Casella, G. (2004). Monte Carlo Statistical Methods, 2nd ed. Springer, New York.
  • Rousseau, J. and Mengersen, K. (2011). Asymptotic behaviour of the posterior distribution in overfitted mixture models. J. R. Stat. Soc. Ser. B Stat. Methodol. 73 689–710.
  • Sperrin, M., Jaki, T. and Wit, E. (2010). Probabilistic relabelling strategies for the label switching problem in Bayesian mixture models. Stat. Comput. 20 357–366.
  • Stephens, M. (1997). Bayesian methods for mixtures of normal distributions. Ph.D. thesis, Dept. Statistics, Univ. Oxford.
  • Stephens, M. (2000). Bayesian analysis of mixture models with an unknown number of components—an alternative to reversible jump methods. Ann. Statist. 28 40–74.
  • Strickland, C. M., Simpson, D. P., Turner, I. W., Denham, R. and Mengersen, K. L. (2011). Fast Bayesian analysis of spatial dynamic factor models for multitemporal remotely sensed imagery. J. R. Stat. Soc. Ser. C. Appl. Stat. 60 109–124.
  • Vesala, T., Hataja, J., Aalto, P., Altimir, N., Buzorius, G., Garam, E., Hämeri, K., Ilvesniemi, H., Jokinen, V., Keronen, P., Lahti, T., Markkanen, T., Mäkelä, J. M., Nikinmaa, E., Palmroth, S., Palva, L., Pohja, T., Pumpanen, J., Rannik, U., Siivola, E., Ylitalo, H., Hari, P. and Kulmala, M. (1998). Long-term field measurements of atmospheric–surface interactions in boreal forest ecology, micrometerology, aerosol physics, and atmospheric chemistry. Trends Heat Mass Momentum Transf. 4 17–35.
  • West, M. and Harrison, J. (1997). Bayesian Forecasting and Dynamic Models, 2nd ed. Springer, New York.
  • Whitby, E., McMurry, P. H. and Shanker, U. (1991). Modal aerosol dynamics modelling. Technical report, US Environment Protection Agency, Atmospheric Research and Exposure Assessment Laboratory.
  • Whitby, E. and McMurry, P. H. (1997). Modal aerosol dynamics modeling. Aerosol Sci. Technol. 27 673–688.
  • Wraith, D., Mengersen, K., Alston, C., Rousseau, J. and Hussein, T. (2014). Supplement to “Using informative priors in the estimation of mixtures over time with application to aerosol particle size distributions.” DOI:10.1214/13-AOAS678SUPP.
  • Yao, W. (2012). Model based labeling for mixture models. Stat. Comput. 22 337–347.
  • World Health Organization (2006). WHO Air quality guidelines for particulate matter, ozone, nitrogen dioxide and sulfur dioxide Global update 2005, World Health Organisation.

Supplemental materials