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Automatic Tempered Posterior Distributions for Bayesian Inversion Problems

Author

Listed:
  • Luca Martino

    (Department of Signal Processing, Universidad rey Juan Carlos (URJC), 28942 Madrid, Spain)

  • Fernando Llorente

    (Department of Statistics, Universidad Carlos III de Madrid (UC3M), 28911 Madrid, Spain)

  • Ernesto Curbelo

    (Department of Statistics, Universidad Carlos III de Madrid (UC3M), 28911 Madrid, Spain)

  • Javier López-Santiago

    (Department of Signal Processing, Universidad Carlos III de Madrid (UC3M), 28911 Madrid, Spain)

  • Joaquín Míguez

    (Department of Signal Processing, Universidad Carlos III de Madrid (UC3M), 28911 Madrid, Spain)

Abstract

We propose a novel adaptive importance sampling scheme for Bayesian inversion problems where the inference of the variables of interest and the power of the data noise are carried out using distinct (but interacting) methods. More specifically, we consider a Bayesian analysis for the variables of interest (i.e., the parameters of the model to invert), whereas we employ a maximum likelihood approach for the estimation of the noise power. The whole technique is implemented by means of an iterative procedure with alternating sampling and optimization steps. Moreover, the noise power is also used as a tempered parameter for the posterior distribution of the the variables of interest. Therefore, a sequence of tempered posterior densities is generated, where the tempered parameter is automatically selected according to the current estimate of the noise power. A complete Bayesian study over the model parameters and the scale parameter can also be performed. Numerical experiments show the benefits of the proposed approach.

Suggested Citation

  • Luca Martino & Fernando Llorente & Ernesto Curbelo & Javier López-Santiago & Joaquín Míguez, 2021. "Automatic Tempered Posterior Distributions for Bayesian Inversion Problems," Mathematics, MDPI, vol. 9(7), pages 1-17, April.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:7:p:784-:d:530560
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    References listed on IDEAS

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    1. Pierre Del Moral & Arnaud Doucet & Ajay Jasra, 2006. "Sequential Monte Carlo samplers," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 68(3), pages 411-436, June.
    2. N. Friel & A. N. Pettitt, 2008. "Marginal likelihood estimation via power posteriors," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 70(3), pages 589-607, July.
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    Cited by:

    1. Curbelo Benitez, Ernesto Angel & Martino, Luca & Llorente Fernandez, Fernando, 2023. "Adaptive posterior distributions for covariance matrix learning in Bayesian inversion problems for multioutput signals," DES - Working Papers. Statistics and Econometrics. WS 37391, Universidad Carlos III de Madrid. Departamento de Estadística.

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