Editing Bayesian inference (section)

===Estimates of parameters and predictions===
It is often desired to use a posterior distribution to estimate a parameter or variable. Several methods of Bayesian estimation select [[central tendency|measurements of central tendency]] from the posterior distribution.

For one-dimensional problems, a unique median exists for practical continuous problems. The posterior median is attractive as a [[robust statistics|robust estimator]].<ref>{{cite book|title=Pitman's measure of closeness: A comparison of statistical estimators|first1=Pranab K.|last1=Sen|author-link1=Pranab K. Sen|first2=J. P.|last2=Keating|first3=R. L.|last3= Mason | publisher=SIAM|location=Philadelphia|year=1993}}</ref>

If there exists a finite mean for the posterior distribution, then the posterior mean is a method of estimation.<ref>{{Cite book| last1=Choudhuri|first1=Nidhan|last2=Ghosal|first2=Subhashis|last3=Roy|first3=Anindya|date=2005-01-01|chapter=Bayesian Methods for Function Estimation|title=Handbook of Statistics|series=Bayesian Thinking|volume=25|pages=373–414|doi= 10.1016/s0169-7161(05)25013-7 |isbn=9780444515391|citeseerx=10.1.1.324.3052}}</ref>
<math display="block">\tilde \theta = \operatorname{E}[\theta] = \int \theta \, p(\theta \mid \mathbf{X},\alpha) \, d\theta</math>

Taking a value with the greatest probability defines [[maximum a posteriori estimation|maximum ''a&nbsp;posteriori'' (MAP)]] estimates:<ref>{{Cite web|url=https://www.probabilitycourse.com/chapter9/9_1_2_MAP_estimation.php|title=Maximum A Posteriori (MAP) Estimation|website=www.probabilitycourse.com|language=en|access-date=2017-06-02}}</ref>
<math display="block">\{ \theta_{\text{MAP}}\} \subset \arg \max_\theta p(\theta \mid \mathbf{X},\alpha) .</math>

There are examples where no maximum is attained, in which case the set of MAP estimates is [[empty set|empty]].

There are other methods of estimation that minimize the posterior ''[[risk]]'' (expected-posterior loss) with respect to a [[loss function]], and these are of interest to [[statistical decision theory]] using the sampling distribution ("frequentist statistics").<ref>{{Cite web|url=http://www.cogsci.ucsd.edu/~ajyu/Teaching/Tutorials/bayes_dt.pdf|title=Introduction to Bayesian Decision Theory|last=Yu|first=Angela|website=cogsci.ucsd.edu/|archive-url=https://web.archive.org/web/20130228060536/http://www.cogsci.ucsd.edu/~ajyu/Teaching/Tutorials/bayes_dt.pdf|archive-date=2013-02-28|url-status=dead}}</ref>

The [[posterior predictive distribution]] of a new observation <math>\tilde{x}</math> (that is independent of previous observations) is determined by<ref>{{Cite web|url=http://people.stat.sc.edu/Hitchcock/stat535slidesday18.pdf|title=Posterior Predictive Distribution Stat Slide|last=Hitchcock|first=David|website=stat.sc.edu}}</ref>
<math display="block">p(\tilde{x}|\mathbf{X},\alpha) = \int p(\tilde{x},\theta \mid \mathbf{X},\alpha) \, d\theta = \int p(\tilde{x} \mid \theta) p(\theta \mid \mathbf{X},\alpha) \, d\theta .</math>