Bayesian Statistics


Bayesian statistics is an approach to data analysis that treats unknown quantities probabilistically, even if the unknowns are fixed (e.g., population parameters) rather than random (e.g., data). The Bayesian approach requires prior probability distributions for such quantities based on what is known about them before observing new data and provides a recipe for updating these priors with new data (a likelihood function) to obtain a posterior distribution that reflects what is known after observing the new data. Thus, a Bayesian analysis involves constructing a prior distribution, constructing a likelihood function for new data, obtaining a posterior distribution, and then summarizing the posterior distribution. Historically, summarizing the posterior distribution involved approximation methods and numerical methods. Contemporary Bayesian applications involve sampling from posterior distributions using Markov chain Monte Carlo (MCMC) methods and then producing basic statistical summary measures, including posterior means, medians, variances, and quantiles. This entry discusses the process of Bayesian analysis from construction of a prior and derivation of a posterior distribution through MCMC development and implementation, to summarizing MCMC output. Advantages of the Bayesian approach using MCMC methods are discussed, including the ease of use of Bayesian methods for contemporary research questions and data, the flexibility the Bayesian approach provides for model evaluation, and the simplicity of the Bayesian approach using MCMC for conducting tertiary analyses of quantities not directly modeled.

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