Bayesian computational methods

In this chapter, we will first present the most standard computational challenges met in Bayesian Statistics, focussing primarily on mixture estimation and on model choice issues, and then relate these problems with computational solutions. Of course, this chapter is only a terse introduction to the problems and solutions related to Bayesian computations. For more complete references, see Robert and Casella (2004, 2009), or Marin and Robert (2007), among others. We also restrain from providing an introduction to Bayesian Statistics per se and for comprehensive coverage, address the reader to Robert (2007), (again) among others.Comment: This is a revised version of a chapter written for the Handbook of Computational Statistics, edited by J. Gentle, W. Hardle and Y. Mori in 2003, in preparation for the second editio

Approximate Bayesian Computational methods

Also known as likelihood-free methods, approximate Bayesian computational (ABC) methods have appeared in the past ten years as the most satisfactory approach to untractable likelihood problems, first in genetics then in a broader spectrum of applications. However, these methods suffer to some degree from calibration difficulties that make them rather volatile in their implementation and thus render them suspicious to the users of more traditional Monte Carlo methods. In this survey, we study the various improvements and extensions made to the original ABC algorithm over the recent years.Comment: 7 figure

Bayesian computational methods

If, in the mid 1980?s, one had asked the average statistician about the difficulties of using Bayesian Statistics, his/her most likely answer would have been ?Well, there is this problem of selecting a prior distribution and then, even if one agrees on the prior, the whole Bayesian inference is simply impossible to implement in practice!? The same question asked in the 21th Century does not produce the same reply, but rather a much less serious complaint about the lack of generic software (besides winBUGS)! The last 15 years have indeed seen a tremendous change in the way Bayesian Statistics are perceived, both by mathematical statisticians and by applied statisticians and the impetus behind this change has been a prodigious leap-forward in the computational abilities. The availability of very powerful approximation methods has correlatively freed Bayesian modelling, in terms of both model scope and prior modelling. As discussed below, a most successful illustration of this gained freedom can be seen in Bayesian model choice, which was only emerging at the beginning of the MCMC era, for lack of appropriate computational tools. In this chapter, we will first present the most standard computational challenges met in Bayesian Statistics (Section 2), and then relate these problems with computational solutions. Of course, this chapter is only a terse introduction to the problems and solutions related to Bayesian computations. For more complete references, see Robert and Casella (1999, 2004) and Liu (2001), among others. We also restrain from providing an introduction to Bayesian Statistics per se and for comprehensive coverage, address the reader to Robert (2001), (again) among others. --

Bayesian Methods for Exoplanet Science

Exoplanet research is carried out at the limits of the capabilities of current telescopes and instruments. The studied signals are weak, and often embedded in complex systematics from instrumental, telluric, and astrophysical sources. Combining repeated observations of periodic events, simultaneous observations with multiple telescopes, different observation techniques, and existing information from theory and prior research can help to disentangle the systematics from the planetary signals, and offers synergistic advantages over analysing observations separately. Bayesian inference provides a self-consistent statistical framework that addresses both the necessity for complex systematics models, and the need to combine prior information and heterogeneous observations. This chapter offers a brief introduction to Bayesian inference in the context of exoplanet research, with focus on time series analysis, and finishes with an overview of a set of freely available programming libraries.Comment: Invited revie

On computational tools for Bayesian data analysis

While Robert and Rousseau (2010) addressed the foundational aspects of Bayesian analysis, the current chapter details its practical aspects through a review of the computational methods available for approximating Bayesian procedures. Recent innovations like Monte Carlo Markov chain, sequential Monte Carlo methods and more recently Approximate Bayesian Computation techniques have considerably increased the potential for Bayesian applications and they have also opened new avenues for Bayesian inference, first and foremost Bayesian model choice.Comment: This is a chapter for the book "Bayesian Methods and Expert Elicitation" edited by Klaus Bocker, 23 pages, 9 figure