A Bayesian Motivated Laplace Inversion for Multivariate Probability Distributions

  • Lorenzo Cappello
  • Stephen G. Walker


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.


Fixed-point Inverse method Recursive estimation Stochastic approximation 

Mathematics Subject Classification (2010)

44A10 62L20 65WP1 


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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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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of Decision SciencesUniversita Commerciale Luigi Bocconi Via Roentgen 1MilanItaly
  2. 2.Department of MathematicsUniversity of Texas at AustinAustinUSA

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