Editing Point estimation (section)

== Types of point estimation ==

=== Bayesian point estimation ===
Bayesian inference is typically based on the [[posterior distribution]]. Many [[Bayesian estimation|Bayesian point estimators]] are the posterior distribution's statistics of [[central tendency]], e.g., its mean, median, or mode:
* [[Bayes estimator#Posterior mean|Posterior mean]], which minimizes the (posterior) [[risk function|''risk'']] (expected loss) for a [[Minimum mean square error|squared-error]] [[loss function]]; in Bayesian estimation, the risk is defined in terms of the posterior distribution, as observed by [[Gauss]].<ref name="Dodge">{{cite book|title=Statistical data analysis based on the L1-norm and related methods: Papers from the First International Conference held at Neuchâtel, August 31–September 4, 1987|publisher=[[North-Holland Publishing]]|year=1987|editor-last=Dodge|editor-first=Yadolah|editor-link=Yadolah Dodge}}</ref>
* [[Bayes estimator#Posterior median and other quantiles|Posterior median]], which minimizes the posterior risk for the absolute-value loss function, as observed by [[Laplace]].<ref name="Dodge" /><ref>{{cite book|last1=Jaynes|first1=E. T.|title=Probability Theory: The logic of science|date=2007|publisher=[[Cambridge University Press]]|isbn=978-0-521-59271-0|edition=5. print.|page=172|author-link=Edwin Thompson Jaynes}}</ref>
* [[maximum a posteriori]] (''MAP''), which finds a maximum of the posterior distribution; for a uniform prior probability, the MAP estimator coincides with the maximum-likelihood estimator;
The MAP estimator has good asymptotic properties, even for many difficult problems, on which the maximum-likelihood estimator has difficulties.
For regular problems, where the maximum-likelihood estimator is consistent, the maximum-likelihood estimator ultimately agrees with the MAP estimator.<ref>{{cite book|last=Ferguson|first=Thomas S.|title=A Course in Large Sample Theory|publisher=[[Chapman & Hall]]|year=1996|isbn=0-412-04371-8|author-link=Thomas S. Ferguson}}</ref><ref name="LeCam">{{cite book|last=Le Cam|first=Lucien|title=Asymptotic Methods in Statistical Decision Theory|publisher=[[Springer-Verlag]]|year=1986|isbn=0-387-96307-3|author-link=Lucien Le Cam}}</ref><ref name="FergJASA">{{cite journal|last=Ferguson|first=Thomas&nbsp;S.|author-link=Thomas S. Ferguson|year=1982|title=An inconsistent maximum likelihood estimate|journal=[[Journal of the American Statistical Association]]|volume=77|issue=380|pages=831–834|doi=10.1080/01621459.1982.10477894|jstor=2287314}}</ref>
Bayesian estimators are [[admissible procedure|admissible]], by Wald's theorem.<ref name="LeCam" /><ref name="LehmannCasella">{{cite book|last1=Lehmann|first1=E. L.|title=Theory of Point Estimation|last2=Casella|first2=G.|publisher=Springer|year=1998|isbn=0-387-98502-6|edition=2nd|author-link=Erich Leo Lehmann}}</ref>

The [[Minimum Message Length]] ([[Minimum Message Length|MML]]) point estimator is based in Bayesian [[information theory]] and is not so directly related to the [[posterior distribution]].

Special cases of [[Bayes filter|Bayesian filters]] are important:
*[[Kalman filter]]
*[[Wiener filter]]

Several [[iterative method|methods]] of [[computational statistics]] have close connections with Bayesian analysis:
*[[particle filter]]
*[[Markov chain Monte Carlo]] (MCMC)