Editing Bayes' theorem (section)

====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>