Posterioronly

Posterioronly is a term used in Bayesian statistics to describe analyses that rely primarily on the posterior distribution of model parameters, obtained by combining a prior with a likelihood via Bayes' theorem. In a posterioronly approach, inferences about quantities of interest are drawn from the posterior, with the prior serving only to form that posterior rather than being considered as an object of separate interpretation.

In practice, posterioronly analyses include computing credible intervals from the posterior, making decisions based on posterior probabilities or posterior risks, and performing model checks that use samples from the posterior predictive distribution. A common tool is the posterior predictive check, where one generates data under the posterior and compares to observed data. Some discussions distinguish posterioronly checks from prior predictive checks, which assess model adequacy by simulating data from the prior before observing data.

Posterioronly methods are central to Bayesian inference, but they come with caveats. Because the posterior mixes prior information with observed data, the resulting inferences can be sensitive to the choice of prior, especially with limited data. Critics note that a purely posterior-centered view may obscure how prior assumptions influence conclusions, while proponents argue that focusing on the posterior aligns with the decision-theoretic aim of Bayesian analysis.

Related concepts include the posterior distribution, the posterior predictive distribution, and Bayesian decision theory. In practice, computational methods such as Markov chain Monte Carlo or variational inference are used to obtain samples from the posterior that underpin posterioronly inference.