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Compound Dirichlet Processes

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Selected Contributions on Statistics and Data Science in Latin America (FNE 2018)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 301))

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Abstract

The compound Poisson process and the Dirichlet process are the pillar structures of renewal theory and Bayesian nonparametric theory, respectively. Both processes have many useful extensions to fulfill the practitioners’ needs to model the particularities of data structures. Accordingly, in this contribution, we join their primal ideas to construct the compound Dirichlet process and the compound Dirichlet process mixture. As a consequence, these new processes have a rich structure to model the time occurrence among events, with also a flexible structure on the arrival variables. These models have a direct Bayesian interpretation of their posterior estimators and are easy to implement. We obtain expressions of posterior distribution, nonconditional distribution, and expected values. In particular, to find these formulas, we analyze sums of random variables with Dirichlet process priors. We assess our approach by applying our model on a real data example of a contagious zoonotic disease.

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Acknowledgements

We thank the editor and two anonymous reviewers for their useful comments which significantly improved the presentation and quality of the paper. The first author is grateful to Prof. Ramsés Mena for the valuable suggestions on an earlier version of the manuscript. This research was partially supported by a DGAPA Postdoctoral Scholarship.

Author information

Authors and Affiliations

  1. Facultad de Ciencias, Departamento de Matemáticas, Universidad Nacional Autónoma de México, Apartado Postal 20-726, 01000, Mexico, CDMX, Mexico

    Arrigo Coen

  2. Departamento de Sistemas Biológicos, División de Ciencias Biológicas y de la Salud, Universidad Autónoma Metropolitana–Xochimilco, Mexico City, Mexico

    Beatriz Godínez-Chaparro

Authors
  1. Arrigo Coen
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  2. Beatriz Godínez-Chaparro
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Corresponding author

Correspondence to Arrigo Coen .

Editor information

Editors and Affiliations

  1. Department of Environmental Sciences, Informatics and Statistics, Ca' Foscari University of Venice, Venice, Italy

    Isadora Antoniano-Villalobos

  2. Department of Probability and Statistics, IIMAS, UNAM, Mexico City, Mexico

    Ramsés H. Mena

  3. Department of Statistics, Instituto Tecnológico Autónomo de México, Mexico City, Mexico

    Manuel Mendoza

  4. Department of Mathematics, Facultad de Ciencias, UNAM, Mexico City, Mexico

    Lizbeth Naranjo

  5. Department of Statistics, Instituto Tecnológico Autónomo de México, Mexico City, Mexico

    Luis E. Nieto-Barajas

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Coen, A., Godínez-Chaparro, B. (2019). Compound Dirichlet Processes. In: Antoniano-Villalobos, I., Mena, R., Mendoza, M., Naranjo, L., Nieto-Barajas, L. (eds) Selected Contributions on Statistics and Data Science in Latin America. FNE 2018. Springer Proceedings in Mathematics & Statistics, vol 301. Springer, Cham. https://doi.org/10.1007/978-3-030-31551-1_4

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