Bayesian statistics and modelling

Bayesian statistics is an approach to data analysis based on Bayes’ theorem, where available knowledge about parameters in a statistical model is updated with the information in observed data. The background knowledge is expressed as a prior distribution and combined with observational data in the form of a likelihood function to determine the posterior distribution. The posterior can also be used for making predictions about future events. This Primer describes the stages involved in Bayesian analysis, from specifying the prior and data models to deriving inference, model checking and refinement. We discuss the importance of prior and posterior predictive checking, selecting a proper technique for sampling from a posterior distribution, variational inference and variable selection. Examples of successful applications of Bayesian analysis across various research fields are provided, including in social sciences, ecology, genetics, medicine and more. We propose strategies for reproducibility and reporting standards, outlining an updated WAMBS (when to Worry and how to Avoid the Misuse of Bayesian Statistics) checklist. Finally, we outline the impact of Bayesian analysis on artificial intelligence, a major goal in the next decade.

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Acknowledgements

R.v.d.S. was supported by grant NWO-VIDI-452-14-006 from the Netherlands Organization for Scientific Research. R.K. was supported by Leverhulme research fellowship grant reference RF-2019-299 and by The Alan Turing Institute under the EPSRC grant EP/N510129/1. K.M. was supported by a UK Engineering and Physical Sciences Research Council Doctoral Studentship. C.Y. is supported by a UK Medical Research Council Research Grant (Ref. MR/P02646X/1) and by The Alan Turing Institute under the EPSRC grant EP/N510129/1