Posteriorlike

Posteriorlike is a term used in statistics to describe distributions or inferences that resemble the Bayesian posterior distribution in shape or interpretation. It is not a formal technical term with a universally agreed definition; rather, it describes objects that perform a similar role to a posterior: summarizing uncertainty about a parameter after observing data and, in many cases, incorporating prior information or prior-like beliefs.

Posteriorlike objects can arise as exact posteriors in fully Bayesian models, but they also appear as approximations or alternative constructions that do not require a full probabilistic model for all data. Common sources include Laplace approximations (Gaussian approximations to the posterior around the mode), variational inference (tractable distributions that minimize divergence to the posterior), bootstrap posteriors (uncertainty estimates derived from resampling), and empirical Bayes or fiducial approaches that yield distributions with posterior-like interpretation.

Usage and considerations: Posteriorlike summaries are valuable when exact Bayesian computation is infeasible or when a decision-maker prefers a distribution with a posterior-like interpretation but within different modeling assumptions. It is important to assess calibration, coverage of credible-like intervals, and sensitivity to the assumed prior or updating mechanism, and to be clear about the extent to which the object can be interpreted as a true posterior.