Regular conditional probability: Difference between revisions

Revision as of 17:37, 8 January 2021

Revision as of 19:30, 8 January 2021

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:<math>P(Y \in A | X = x) = \frac{\int_A f_{X,Y}(x, y) \mathrm{d}y}{\int_\mathbb{R} f_{X,Y}(x, y) \mathrm{d}y}.</math>

Conditional probability is a two variable function as before, undefined outside of the [[support]] of the distribution of ''X''.

==Relation to conditional expectation==

In probability theory, the theory of [[conditional expectation]] is developed before that of regular conditional distributions.<ref>{{cite book |last1=Durrett |first1=Richard |title=Probability : theory and examples |date=2010 |publisher=Cambridge University Press |___location=Cambridge |isbn=9780521765398 |edition=4th}}</ref><ref>{{cite book |last1=Klenke |first1=Achim |title=Probability theory : a comprehensive course |___location=London |isbn=978-1-4471-5361-0 |edition=Second}}</ref>

For discrete random variables and continuous random variables, the conditional expectation is given by

:<math>

\begin{aligned}

\mathbb{E}[X|Y=y] &= \sum_x x P(X=x|Y=y)\\

\mathbb{E}[X|Y=y] &= \int x f_{X|Y}(x, y) \mathrm{d}x

\end{aligned}

</math>

where <math>f_{X|Y}(x, y)</math> is the [[conditional density]] of {{mvar|X}} given {{mvar|Y}}.

It is natural to ask whether measure theoretical conditional expectation can also be expressed as

:<math>\mathbb{E}[X|Y](\omega) = \int x \nu(\omega, \mathrm{d}x)</math>

where <math>\nu : \Omega \times \mathcal{B}(\overline{\mathbb{R}}) \to [0,1]</math> is a family of measures parametrized by outcome <math>\omega</math>.

Such a [[Markov kernel]] can be defined using conditional expectation:

:<math>\nu(\omega, A) = \mathbb{E}[1_{X \in A} | Y](\omega)</math>

It can be shown that for almost all <math>\omega</math>, this is a probability measure if <math>X : \Omega \to \mathbb{R}</math>. There are, however, counterexamples when the random variable {{mvar|X}} takes values in a more general space {{mvar|E}}. A space {{mvar|E}} can be constructed where <math>\nu</math> does not form a probability measure almost everywhere.

==Definition==