Markov chain Monte Carlo–based Bayesian data analysis has now become the method of choice for analyzing and interpreting data in almost all disciplines of science. In astronomy, over the past decade, we have also seen a steady increase in the number of papers that employ Monte Carlo–based Bayesian analysis. New, efficient Monte Carlo–based methods are continuously being developed and explored. In this review, we first explain the basics of Bayesian theory and discuss how to set up data analysis problems within this framework. Next, we provide an overview of various Monte Carlo–based methods for performing Bayesian data analysis. Finally, we discuss advanced ideas that enable us to tackle complex problems and thus hold great promise for the future. We also distribute downloadable computer software () Python that implements some of the algorithms and examples discussed here.


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