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In this important new Handbook, the editors have gathered together a range of leading contributors to introduce the theory and practice of multilevel modeling.

The Handbook establishes the connections in multilevel modeling, bringing together leading experts from around the world to provide a roadmap for applied researchers linking theory and practice, as well as a unique arsenal of state-of-the-art tools. It forges vital connections that cross traditional disciplinary divides and introduces best practice in the field.

Part I establishes the framework for estimation and inference, including chapters dedicated to notation, model selection, fixed and random effects, and causal inference; Part II develops variations and extensions, such as nonlinear, semiparametric and latent class models; Part III includes discussion of missing data and robust methods, assessment of fit and ...

Bayesian Multilevel Models
Bayesian multilevel models
LudwigFahrmeirUniversity of Munich (Emeritus), Germany
ThomasKneibUniversity of Göttingen, Germany
StefanLangUniversity of Innsbruck, Austria
4.1 Bayesian Inference in a Nutshell

This section is a strongly condensed version of an introduction to Bayesian inference given in Fahrmeir et al. (2013) Appendix B. Other, more elaborate, introductions to Bayesian inference are presented, e.g. in O'Hagan (1994), Gelman et al. (2003), and Held and Sabanes-Bove (2012).

The fundamental difference to likelihood-based inference is that the unknown parameters in a statistical model, θ = (θ1,…, θp)’, say, are not considered as fixed, deterministic quantities but as variables with a prior distribution. A Bayesian model therefore consists of two parts:

  • Prior distribution: Any (possibly subjective) information about the unknown parameter θ is expressed by specifying a probability distribution for θ. This distribution ...
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