Skip to main content

A Bayesian Motivated Laplace Inversion for Multivariate Probability Distributions


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.

This is a preview of subscription content, access via your institution.


  • Abate J, Choudhury GL, Whitt W (1998) Numerical inversion of multidimensional Laplace transforms by the Laguerre method. Perform Eval 31 (3):229–243

    Article  Google Scholar 

  • Abate J, Whitt W (1992) The Fourier-series method for inverting transforms of probability distributions. Queueing Systems 10(1):5–87

    MathSciNet  Article  MATH  Google Scholar 

  • Abate J, Whitt W (2006) A unified framework for numerically inverting Laplace transforms. INFORMS J Comput 18(4):408–421

    MathSciNet  Article  MATH  Google Scholar 

  • Asmussen S, Jensen JL, Rojas-Nandayapa L (2016) On the Laplace transform of the lognormal distribution. Methodol Comput Appl Probab 18(2):441–458

    MathSciNet  Article  MATH  Google Scholar 

  • Billingsley P (1999) Convergence of probability measures. Wiley, Chicago

    Book  MATH  Google Scholar 

  • Caflisch RE (1998) Monte Carlo and Quasi-Monte Carlo methods. Acta Numerica 7(1):1–49

    MathSciNet  Article  MATH  Google Scholar 

  • Cai N, Kou S (2012) Pricing Asian options under a hyper-exponential jump diffusion model. Oper Res 60(1):64–77

    MathSciNet  Article  MATH  Google Scholar 

  • Choudhury GL, Lucantoni DM, Whitt W (1994) Multidimensional transform inversion with applications to the transient m/g/1 queue. Ann Appl Probab 4(3):719–740

    MathSciNet  Article  MATH  Google Scholar 

  • Downton F (1970) Bivariate exponential distributions in reliability theory. J R Stat Soc Ser B Methodol 32(3):408–417

    MathSciNet  MATH  Google Scholar 

  • Dubner H, Abate J (1968) Numerical inversion of Laplace transforms by relating them to the finite Fourier cosine transform. J ACM (JACM) 15(1):115–123

    MathSciNet  Article  MATH  Google Scholar 

  • Feller WG (1971) An introduction to probability theory and its applications, vol 2. Wiley, New York

    MATH  Google Scholar 

  • Fusai G (2004) Pricing Asian options via Fourier and Laplace transforms. Journal of Computational Finance 7(3):87–106

    Article  Google Scholar 

  • Gaver Jr DP (1966) Observing stochastic processes, and approximate transform inversion. Oper Res 14(3):444–459

    MathSciNet  Article  Google Scholar 

  • Gelenbe E (2006) Computer system performance modeling in perspective: a tribute to the work of professor Kenneth C. Sevcik, vol 1. Imperial College Press, London

    Book  MATH  Google Scholar 

  • Goffard P-O, Loisel S, Pommeret D (2015) Polynomial approximations for bivariate aggregate claims amount probability distributions. Methodol Comput Appl Probab 19(1):151–174

    MathSciNet  Article  MATH  Google Scholar 

  • Grübel R, Hermesmeier R (1999) Computation of compound distributions I: Aliasing errors and exponential tilting. Astin Bull. 29(2):197–214

    Article  Google Scholar 

  • Hammersley JM, Handscomb DC (1964) General principles of the Monte Carlo method. In: Monte Carlo Methods. Monographs on Applied Probability and Statistics. Springer, Dordrecht

  • Jin T, Provost SB, Ren J (2016) Moment-based density approximations for aggregate losses. Scand Actuar J 2016(3):216–245

    MathSciNet  Article  Google Scholar 

  • Jin T, Ren J (2010) Recursions and fast Fourier transforms for certain bivariate compound distributions. Journal of Operational Risk 5(4):19–33

    Article  Google Scholar 

  • Kuznetsov A (2013) On the convergence of the Gaver–Stehfest algorithm. SIAM J Numer Anal 51(6):2984–2998

    MathSciNet  Article  MATH  Google Scholar 

  • Mnatsakanov R (2011) Moment-recovered approximations of multivariate distributions: the Laplace transform inversion. Statistics & probability letters 81(1):1–7

    MathSciNet  Article  MATH  Google Scholar 

  • Mnatsakanov R, Ruymgaart F (2003) Some properties of moment-empirical cdf’s with application to some inverse estimation problems. Mathematical Methods of Statistics 12(4):478–495

    MathSciNet  Google Scholar 

  • Morokoff WJ, Caflisch RE (1995) Quasi-monte Carlo integration. J Comput Phys 122(2):218–230

    MathSciNet  Article  MATH  Google Scholar 

  • Morris CN (1982) Natural exponential families with quadratic variance functions. Ann Stat 10(1):65–80

    MathSciNet  Article  MATH  Google Scholar 

  • Mustapha H, Dimitrakopoulos R (2010) Generalized Laguerre expansions of multivariate probability densities with moments. Computers & Mathematics with Applications 60(7):2178–2189

    MathSciNet  Article  MATH  Google Scholar 

  • Newton MA, Quintana FA, Zhang Y (1998) Nonparametric Bayes methods using predictive updating. In: Practical nonparametric and semiparametric bayesian statistics, vol 133. Springer, New York, pp 45–61

  • Niederreiter H (1992) Random Number Generation and Quasi-Monte Carlo Methods Number 63 in CBMS-NSF Series in Applied Mathematics. SIAM, Philadelphia

  • Ridout M (2009) Generating random numbers from a distribution specified by its Laplace transform. Stat Comput 19(4):439–450

    MathSciNet  Article  Google Scholar 

  • Rojas-Nandayapa L (2008) Risk probabilities: asymptotics and simulation. Ph.D. thesis, Aarhus University

  • Stehfest H (1970) Algorithm 368: Numerical inversion of Laplace transforms. Commun ACM 13(1):47–49

    Article  Google Scholar 

  • Talbot A (1979) The accurate numerical inversion of Laplace transforms. IMA J Appl Math 23(1):97–120

    MathSciNet  Article  MATH  Google Scholar 

  • Walker SG (2017) A Laplace transform inversion method for probability functions. Stat Comput 27(2):439–448

    MathSciNet  Article  MATH  Google Scholar 

  • Weideman JAC (2006) Optimizing Talbot’s contours for the inversion of the Laplace transform. SIAM J Numer Anal 44(6):2342–2362

    MathSciNet  Article  MATH  Google Scholar 

Download references


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.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Lorenzo Cappello.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Cappello, L., Walker, S.G. A Bayesian Motivated Laplace Inversion for Multivariate Probability Distributions. Methodol Comput Appl Probab 20, 777–797 (2018).

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


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

Mathematics Subject Classification (2010)

  • 44A10
  • 62L20
  • 65WP1