Bayesian Inference

This chapter provides a overview of Bayesian inference, mostly emphasising that it is auniversal method for summarising uncertainty and making estimates and predictions usingprobability statements conditional on observed data and an assumed model (Gelman 2008).The Bayesian perspective is thus applicable to all aspects of statistical inference, while beingopen to the incorporation of information items resulting from earlier experiments and fromexpert opinions.We provide here the basic elements of Bayesian analysis when considered for standardmodels, refering to Marin and Robert (2007) and to Robert (2007) for book-length entries.1In the following, we refrain from embarking upon philosophical discussions about the natureof knowledge (see, e.g., Robert 2007, Chapter 10), opting instead for a mathematically soundpresentation of an eminently practical statistical methodology. We indeed believe that themost convincing arguments for adopting a Bayesian version of data analyses are in theversatility of this tool and in the large range of existing applications, rather than in thosepolemical arguments (for such perspectives, see, e.g., Jaynes 2003 and MacKay 2002).