Regular conditional distribution

The regular conditional distribution of a random variable is a term from probability theory . It generalizes the distribution of a random variable around the aspect that previous information about the possible outcomes of a random experiment may already be known. The regular conditional distribution thus plays an important role in Bayesian statistics and in the theory of stochastic processes . In contrast to the (ordinary) conditional distribution , the regular conditional distribution is defined with the help of the conditional expectation and not with the (ordinary) conditional probability , which makes it much more general.

definition

A probability space and a measurement space as well as a sub-σ-algebra of are given . Let be a random variable from to . ${\displaystyle (\Omega ,{\mathcal {A}},P)}$ ${\displaystyle (E,{\mathcal {E}})}$ ${\displaystyle {\mathcal {F}}}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle Y}$ ${\displaystyle (\Omega ,{\mathcal {A}})}$ ${\displaystyle (E,{\mathcal {E}})}$

A Markov kernel from to is called a regular version of the conditional distribution of the random variable given if ${\displaystyle \kappa _{Y,{\mathcal {F}}}}$ ${\displaystyle (\Omega ,{\mathcal {A}})}$ ${\displaystyle (E,{\mathcal {E}})}$ ${\displaystyle Y}$ ${\displaystyle {\mathcal {F}}}$

${\displaystyle \kappa _{Y,{\mathcal {F}}}(\omega ,B)=P(Y^{-1}(B)|{\mathcal {F}})(\omega )}$

for everyone and for - almost everyone . ${\displaystyle B\in {\mathcal {E}}}$ ${\displaystyle P}$ ${\displaystyle \omega }$

Here is the conditional probability as it is defined by the conditional expected value . ${\displaystyle P(A|{\mathcal {F}})(\omega ):=\operatorname {E} (\mathbf {1} _{A}|{\mathcal {F}})(\omega )}$

The conditions in the definition of the function explicitly mean : ${\displaystyle \kappa _{Y,{\mathcal {F}}}\colon \Omega \times {\mathcal {E}}\to [0,1]}$

1. For all is a probability measure on , ${\displaystyle \omega \in \Omega }$ ${\displaystyle \kappa _{Y,{\mathcal {F}}}(\omega ,\,\cdot \,)}$ ${\displaystyle (E,{\mathcal {E}})}$
2. for everyone is a measurable function and ${\displaystyle B\in {\mathcal {E}}}$ ${\displaystyle \kappa _{Y,{\mathcal {F}}}(\,\cdot \,,B)}$ ${\displaystyle {\mathcal {F}}}$
3. for all and all true . ${\displaystyle B\in {\mathcal {E}}}$ ${\displaystyle F\in {\mathcal {F}}}$ ${\displaystyle \int _{F}\kappa _{Y,{\mathcal {F}}}(\,\cdot \,,B)\,dP=P(Y^{-1}(B)\cap F)}$

Remarks

existence

A regular conditional distribution always exists for real-valued random variables if the real numbers are provided with Borel's σ-algebra . More generally, the regular conditional distribution always exists for random variables with values ​​in Borel spaces , for example for Polish spaces or those provided with Borel's σ-algebra. ${\displaystyle \mathbb {R} ^{n}}$

variants

Analogous to the variants of the conditional expectation value, different variants of the regular conditional distribution can also be defined, all of which can be traced back to the above definition.

• Without the use of random variables, the conditional distribution of given can be defined as the Markov kernel with ${\displaystyle P}$ ${\displaystyle {\mathcal {F}}}$

${\displaystyle \kappa (\omega ,A)=P(A|{\mathcal {F}})(\omega )}$

for - almost everyone and everyone . ${\displaystyle P}$ ${\displaystyle \omega }$ ${\displaystyle A\in {\mathcal {A}}}$

• If another random variable is from in to another measurement space , the σ-algebra is replaced by the σ-algebra generated by the random variable in order to obtain the conditional distribution of given . ${\displaystyle X}$ ${\displaystyle (\Omega ,{\mathcal {A}})}$ ${\displaystyle (E_{1},{\mathcal {E}}_{1})}$ ${\displaystyle {\mathcal {F}}}$ ${\displaystyle X}$ ${\displaystyle \sigma (X)}$ ${\displaystyle Y}$ ${\displaystyle X}$

example

Given are two real-valued random variables with a common density function with respect to the Lebesgue measure. Then the regular conditional distribution is given by the density ${\displaystyle f_{X,Y}(x,y)}$ ${\displaystyle Y}$ ${\displaystyle X}$

${\displaystyle f_{Y|X}(y|x):={\frac {f(x,y)}{f_{X}(x)}}}$,

that means it applies

${\displaystyle \kappa _{Y,\sigma (X)}(\omega ,B)={\frac {\displaystyle \int _{B}f(X(\omega ),y)\,dy}{f_{X}(X(\omega ))}}}$.

Here the density of the marginal distribution denotes . The fact that this marginal distribution can become zero in the denominator is not problematic, since this only happens on a -No set. ${\displaystyle f_{X}(x)=\int _{\mathbb {R} }f(x,y)\,dy}$ ${\displaystyle P_{X}}$

Calculation of conditional expected values

If a regular version of the conditional distribution of an integrable real-valued random variable is given , then for the conditional expectation of given ${\displaystyle \kappa _{Y,{\mathcal {F}}}}$ ${\displaystyle Y}$ ${\displaystyle {\mathcal {F}}}$ ${\displaystyle Y}$ ${\displaystyle {\mathcal {F}}}$

${\displaystyle \operatorname {E} (Y|{\mathcal {F}})(\omega )=\int _{\mathbb {R} }y\,\kappa _{Y,{\mathcal {F}}}(\omega ,dy)}$

for almost everyone . ${\displaystyle P}$ ${\displaystyle \omega \in \Omega }$

literature

• Achim Klenke: Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , doi : 10.1007 / 978-3-642-36018-3 .
• Ludger Rüschendorf: Mathematical Statistics . Springer Verlag, Berlin Heidelberg 2014, ISBN 978-3-642-41996-6 , doi : 10.1007 / 978-3-642-41997-3 .