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International Conference on Bayesian Statistics in Action

BAYSM 2016: Bayesian Statistics in Action pp 11–19Cite as

On the Truncation Error of a Superposed Gamma Process

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Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 194))

Abstract

Completely random measures (CRMs) form a key ingredient of a wealth of stochastic models, in particular in Bayesian Nonparametrics for defining prior distributions. CRMs can be represented as infinite series of weighted random point masses. A constructive representation due to Ferguson and Klass provides the jumps of the series in decreasing order. This feature is of primary interest when it comes to sampling since it minimizes the truncation error for a fixed truncation level of the series. In this paper we focus on a general class of CRMs, namely the superposed gamma process, which suitably transformed has already been successfully implemented in Bayesian Nonparametrics, and quantify the quality of the approximation in two ways. First, we derive a bound in probability for the truncation error. Second, following [1], we study a moment-matching criterion which consists in evaluating a measure of discrepancy between actual moments of the CRM and moments based on the simulation output. To this end, we show that the moments of this class of processes can be obtained analytically.

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Acknowledgements

The authors are grateful to two anonymous referees for valuable comments and suggestions. Julyan Arbel was a postdoc at Bocconi University and Collegio Carlo Alberto, Italy, when this article was submitted. This work is supported by the European Research Council (ERC) through StG “N-BNP” 306406.

Author information

Authors and Affiliations

  1. Inria Grenoble - Rhône-Alpes, Grenoble, France

    Julyan Arbel

  2. Université Grenoble Alpes, Grenoble, France

    Julyan Arbel

  3. Department of Decision Sciences, BIDSA and IGIER, Bocconi University, Milan, Italy

    Igor Prünster

Authors
  1. Julyan Arbel
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  2. Igor Prünster
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Corresponding author

Correspondence to Julyan Arbel .

Editor information

Editors and Affiliations

  1. CNR–IMATI, Milano, Italy

    Raffaele Argiento

  2. CNR-IMATI, Milan, Italy

    Ettore Lanzarone

  3. Decision Sciences, Bocconi University, Milan, Italy

    Isadora Antoniano Villalobos

  4. Dipartimento di Statistica "G. Parenti", Università di Firenze, Firenze, Italy

    Alessandra Mattei

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Arbel, J., Prünster, I. (2017). On the Truncation Error of a Superposed Gamma Process. In: Argiento, R., Lanzarone, E., Antoniano Villalobos, I., Mattei, A. (eds) Bayesian Statistics in Action. BAYSM 2016. Springer Proceedings in Mathematics & Statistics, vol 194. Springer, Cham. https://doi.org/10.1007/978-3-319-54084-9_2

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