Bayesians

Bayesians are adherents of Bayesian statistics and Bayesian epistemology who treat probability as a measure of personal or rational belief and who apply Bayes' theorem to update beliefs in light of new evidence. They focus on incorporating prior information with data to form updated beliefs.

In statistics, Bayesians specify a prior distribution representing beliefs about parameters before observing data, a likelihood function describing the data-generating process, and a posterior distribution that results from combining prior and data. The posterior is used to quantify uncertainty and to make inferences about parameters.

The posterior is used to make inferences, quantify uncertainty, and generate predictive distributions for future observations. Bayesian methods rely on computational tools such as Markov chain Monte Carlo, variational inference, and hierarchical models. They emphasize a coherent probabilistic framework for updating beliefs and for propagating uncertainty through analyses.

Bayesian approaches emphasize model comparison and uncertainty propagation, using techniques like Bayes factors, model averaging, or full probabilistic modeling. Critics argue that prior choices introduce subjectivity, and that computations can be intensive, especially in high dimensions. Proponents argue that priors can be chosen to reflect genuine prior information and that the framework provides a principled way to incorporate uncertainty.

Philosophically, Bayesianism offers a framework for rational belief revision, with different schools advocating subjective or objective priors. It stands in contrast to frequentist statistics, which interprets probability as long-run frequencies and uses different inferential rules.

Historically, the idea traces to Thomas Bayes and was formalized by Pierre-Simon Laplace; Bayesian methods became prominent in the late 20th century with advances in computation and probabilistic programming.

Bayesians are active in fields ranging from science and engineering to machine learning, economics, and philosophy.