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A Bayesian Motivated Laplace Inversion for Multivariate Probability Distributions

Methodology and Computing in Applied Probability volume 20, pages 777–797 (2018)Cite this article

Abstract

The paper introduces a recursive procedure to invert the multivariate Laplace transform of probability distributions. The procedure involves taking independent samples from the Laplace transform; these samples are then used to update recursively an initial starting distribution. The update is Bayesian driven. The final estimate can be written as a mixture of independent gamma distributions, making it the only methodology which guarantees to numerically recover a probability distribution with positive support. Proof of convergence is given by a fixed point argument. The estimator is fast, accurate and can be run in parallel since the target distribution is evaluated on a grid of points. The method is illustrated on several examples and compared to the bivariate Gaver–Stehfest method.

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Acknowledgements

The authors are grateful for the comments and suggestions of two reviewers and the Editor on a previous version of the paper. The first author is partially funded by Cariplo. The second author is partially funded by NSF grant DMS 1612891.

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Authors and Affiliations

  1. Department of Decision Sciences, Universita Commerciale Luigi Bocconi Via Roentgen 1, 20136, Milan, Italy

    Lorenzo Cappello

  2. Department of Mathematics, University of Texas at Austin, 2515 Speedway, Austin, TX, 78712, USA

    Stephen G. Walker

Authors
  1. Lorenzo Cappello
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  2. Stephen G. Walker
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Correspondence to Lorenzo Cappello.

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Cappello, L., Walker, S.G. A Bayesian Motivated Laplace Inversion for Multivariate Probability Distributions. Methodol Comput Appl Probab 20, 777–797 (2018). https://doi.org/10.1007/s11009-017-9587-y

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  • DOI: https://doi.org/10.1007/s11009-017-9587-y

Keywords

  • Fixed-point
  • Inverse method
  • Recursive estimation
  • Stochastic approximation

Mathematics Subject Classification (2010)

  • 44A10
  • 62L20
  • 65WP1