Editing Statistical inference (section)

====Bayesian inference, subjectivity and decision theory====

Many informal Bayesian inferences are based on "intuitively reasonable" summaries of the posterior. For example, the posterior mean, median and mode, highest posterior density intervals, and Bayes Factors can all be motivated in this way. While a user's [[utility function]] need not be stated for this sort of inference, these summaries do all depend (to some extent) on stated prior beliefs, and are generally viewed as subjective conclusions. (Methods of prior construction which do not require external input have been [[Bayesian probability#Personal probabilities and objective methods for constructing priors|proposed]] but not yet fully developed.)

Formally, Bayesian inference is calibrated with reference to an explicitly stated utility, or loss function; the 'Bayes rule' is the one which maximizes expected utility, averaged over the posterior uncertainty. Formal Bayesian inference therefore automatically provides [[optimal decision]]s in a [[decision theory|decision theoretic]] sense. Given assumptions, data and utility, Bayesian inference can be made for essentially any problem, although not every statistical inference need have a Bayesian interpretation. Analyses which are not formally Bayesian can be (logically) [[Coherence (statistics)|incoherent]]; a feature of Bayesian procedures which use proper priors (i.e. those integrable to one) is that they are guaranteed to be [[Coherence (statistics)|coherent]]. Some advocates of [[Bayesian inference]] assert that inference ''must'' take place in this decision-theoretic framework, and that [[Bayesian inference]] should not conclude with the evaluation and summarization of posterior beliefs.