Editing Bayes' theorem (section)

==Statement of theorem==
Bayes' theorem is stated mathematically as the following equation:<ref>{{Citation | first1= A. | last1= Stuart | first2= K. | last2= Ord | title= Kendall's Advanced Theory of Statistics: Volume I – Distribution Theory | year= 1994 | publisher= [[Edward Arnold (publisher)|Edward Arnold]] | at= §8.7}}</ref>

{{Equation box 1
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|equation = <math>P(A\vert B) = \frac{P(B \vert A) P(A)}{P(B)}</math>
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where <math>A</math> and <math>B</math> are [[Event (probability theory)|events]] and <math>P(B) \neq 0</math>.

* <math>P(A\vert B)</math> is a [[conditional probability]]: the probability of event <math>A</math> occurring given that <math>B</math> is true. It is also called the [[posterior probability]] of <math>A</math> given <math>B</math>.
* <math>P(B\vert A)</math> is also a conditional probability: the probability of event <math>B</math> occurring given that <math>A</math> is true. It can also be interpreted as the [[Likelihood function|likelihood]] of <math>A</math> given a fixed <math>B</math> because <math>P(B\vert A)=L(A\vert B)</math>.
* <math>P(A)</math> and <math>P(B)</math> are the probabilities of observing <math>A</math> and <math>B</math> respectively without any given conditions; they are known as the [[prior probability]] and [[marginal probability]].

===Proof===
[[File:Bayes_theorem_visual_proof.svg|thumb|upright|Visual proof of {{nowrap|Bayes' theorem}}]]

====For events====
Bayes' theorem may be derived from the definition of [[conditional probability]]:

:<math>P(A\vert B)=\frac{P(A \cap B)}{P(B)}, \text{ if } P(B) \neq 0, </math>

where <math>P(A \cap B)</math> is the probability of both A and B being true. Similarly,

:<math>P(B\vert A)=\frac{P(A \cap B)}{P(A)}, \text{ if } P(A) \neq 0. </math>

Solving for <math>P(A \cap B)</math> and substituting into the above expression for <math>P(A\vert B)</math> yields Bayes' theorem:

:<math>P(A\vert B) = \frac{P(B\vert A) P(A)}{P(B)}, \text{ if } P(B) \neq 0.</math>

====For continuous random variables====
For two continuous [[random variable]]s ''X'' and ''Y'', Bayes' theorem may be analogously derived from the definition of [[conditional density]]:

:<math>f_{X \vert  Y=y} (x) = \frac{f_{X,Y}(x,y)}{f_Y(y)} </math>
:<math>f_{Y \vert  X=x}(y) = \frac{f_{X,Y}(x,y)}{f_X(x)} </math>

Therefore,

:<math>f_{X \vert Y=y}(x) = \frac{f_{Y \vert  X=x}(y) f_X(x)}{f_Y(y)}.</math>
This holds for values <math>x</math> and <math>y</math> within the [[Support (mathematics)|support]] of ''X'' and ''Y'', ensuring <math>f_X(x) > 0</math> and <math>f_Y(y)>0</math>.

====General case====
Let <math>P_Y^x </math> be the conditional distribution of <math>Y</math> given <math>X = x</math> and let <math>P_X</math> be the distribution of <math>X</math>. The joint distribution is then <math>P_{X,Y} (dx,dy) = P_Y^x (dy) P_X (dx)</math>. The conditional distribution <math>P_X^y </math> of <math>X</math> given <math>Y=y</math> is then determined by

<math display="block">P_X^y (A) = E (1_A (X) | Y = y)</math>

Existence and uniqueness of the needed [[conditional expectation]] is a consequence of the [[Radon–Nikodym theorem]]. This was formulated by [[Andrey Kolmogorov|Kolmogorov]] in 1933. Kolmogorov underlines the importance of conditional probability, writing, "I wish to call attention to ... the theory of conditional probabilities and conditional expectations".<ref>{{Cite book |last=Kolmogorov |first=A.N. |title=Foundations of the Theory of Probability |publisher=Chelsea Publishing Company |orig-year=1956 |year=1933}}</ref> Bayes' theorem determines the posterior distribution from the prior distribution. Uniqueness requires continuity assumptions.<ref>{{Cite book |last=Tjur |first=Tue |url=http://archive.org/details/probabilitybased0000tjur |title=Probability based on Radon measures |date=1980 |location=New York |publisher=Wiley |isbn=978-0-471-27824-5}}</ref> Bayes' theorem can be generalized to include improper prior distributions such as the uniform distribution on the real line.<ref>{{Cite journal |last1=Taraldsen |first1=Gunnar |last2=Tufto |first2=Jarle |last3=Lindqvist |first3=Bo H. |date=2021-07-24 |title=Improper priors and improper posteriors |journal=Scandinavian Journal of Statistics |volume=49 |issue=3 |language=en |pages=969–991 |doi=10.1111/sjos.12550 |s2cid=237736986 |issn=0303-6898|doi-access=free |hdl=11250/2984409 |hdl-access=free }}</ref> Modern [[Markov chain Monte Carlo]] methods have boosted the importance of Bayes' theorem, including in cases with improper priors.<ref>{{Cite book |last1=Robert |first1=Christian P. |url=http://worldcat.org/oclc/1159112760 |title=Monte Carlo Statistical Methods |last2=Casella |first2=George |publisher=Springer |year=2004 |isbn=978-1475741452 |oclc=1159112760}}</ref>