Bayes Theorem: Difference between revisions
imported>Michael Hardy No edit summary |
imported>Michael Hardy ("Likelihood" is an unfortunate word choice, since it has a technical meaning in this context conflicting with this particular use.) |
||
Line 29: | Line 29: | ||
: <math> P(H_1\mid D),\dots,P(H_n\mid D).\, </math> | : <math> P(H_1\mid D),\dots,P(H_n\mid D).\, </math> | ||
In [[epidemiology]], it is used to obtain the probability of disease in a group of people with some characteristic on the basis of the overall rate of that disease and of the | In [[epidemiology]], it is used to obtain the probability of disease in a group of people with some characteristic on the basis of the overall rate of that disease and of the probabilities of that characteristic in healthy and diseased individuals. In clinical decision analysis it is used for estimating the probability of a particular diagnosis given the appearance of some symptoms or test result.<ref name="MeSH">{{cite web |url=http://www.nlm.nih.gov/cgi/mesh/2008/MB_cgi?mode= |title=Bayes Theorem |accessdate=2007-12-09 |author=National Library of Medicine |authorlink= |coauthors= |date= |format= |work= |publisher= |pages= |language= |archiveurl= |archivedate= |quote=}}</ref> | ||
==Calculations== | ==Calculations== |
Revision as of 19:58, 19 December 2007
Bayes' Theorem is a theorem in probability theory named for Thomas Bayes (1702–1761).
It is used for updating probabilities by finding conditional probabilities given new data. This simplest case involves a situation in which probabilities have been assigned to each of several mutually exclusive alternatives H1, ..., Hn, at least one of which may be true. New data D is observed. The conditional probability of D given each of the alternative hypotheses H1, ..., Hn is known. What is needed is the conditional probability of each hypothesis Hi given D. Bayes' Theorem says
The use of Bayes' Theorem is sometimes described as follows. Start with the vector of "prior probabilities", i.e. the probabilities of the several hypotheses before the new data is observed:
Multiply these term-by-term by the "likelihood vector":
getting
The sum of these numbers is not (usually) 1. Multiply all of them by the "normalizing constant"
getting
The result is the "posterior probabilities", i.e. conditional probabilities given the new data:
In epidemiology, it is used to obtain the probability of disease in a group of people with some characteristic on the basis of the overall rate of that disease and of the probabilities of that characteristic in healthy and diseased individuals. In clinical decision analysis it is used for estimating the probability of a particular diagnosis given the appearance of some symptoms or test result.[1]
Calculations
References
- ↑ National Library of Medicine. Bayes Theorem. Retrieved on 2007-12-09.