Bayesian Inference

TL;DR

This chapter offers a comprehensive overview of Bayesian inference, emphasizing its versatility for uncertainty quantification and prediction in statistical analysis, applicable across various models and incorporating prior information.

Contribution

It provides a practical, mathematically sound presentation of Bayesian analysis for standard models, highlighting its broad applicability and utility in statistical inference.

Findings

Bayesian inference is a universal method for uncertainty and prediction.

It effectively incorporates prior information and data.

The approach is versatile across different models.

Abstract

This chapter provides a overview of Bayesian inference, mostly emphasising that it is a universal method for summarising uncertainty and making estimates and predictions using probability statements conditional on observed data and an assumed model (Gelman 2008). The Bayesian perspective is thus applicable to all aspects of statistical inference, while being open to the incorporation of information items resulting from earlier experiments and from expert opinions. We provide here the basic elements of Bayesian analysis when considered for standard models, refering to Marin and Robert (2007) and to Robert (2007) for book-length entries.1 In the following, we refrain from embarking upon philosophical discussions about the nature of knowledge (see, e.g., Robert 2007, Chapter 10), opting instead for a mathematically sound presentation of an eminently practical statistical methodology. We indeed believe that the most convincing arguments for adopting a Bayesian version of data analyses are in the versatility of this tool and in the large range of existing applications, rather than in those polemical arguments.