Conditional independence

In probability theory, conditional independence is a generalization of the stochastic independence of events , set systems and random variables by means of the conditional probability and the conditional expected value . The conditional independence is used, for example, for statements about exchangeable families of random variables .

definition

Let a probability space and a sub- σ-algebra of . Be given the conditional probability . ${\displaystyle (\Omega ,\Sigma ,P)}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle \Sigma }$ ${\displaystyle P(\cdot |{\mathcal {A}})}$ ${\displaystyle {\mathcal {A}}}$

A family of partial σ-algebras of is called conditionally independent if for every finite subset of and every arbitrary choice of with holds that ${\displaystyle ({\mathcal {A}}_{i})_{i\in I}}$ ${\displaystyle \Sigma }$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle J}$ ${\displaystyle I}$ ${\displaystyle A_{j}\in {\mathcal {A}}_{j}}$ ${\displaystyle j\in J}$

${\displaystyle P\left(\bigcap _{j\in J}A_{j}|{\mathcal {A}}\right)=\prod _{j\in J}P(A_{j}|{\mathcal {A}})}$.

Due to the properties of the conditional probability, the identity is to be understood as P-almost certain.

A family of random variables is said to be conditionally given independently if the family of the generated σ-algebras is conditionally given independently . ${\displaystyle (X_{i})_{i\in I}}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle (\sigma (X_{i}))_{i\in I}}$ ${\displaystyle {\mathcal {A}}}$

Comments and characteristics

• Based on the phrase " independently identically distributed ", the conditional probability is used to define a family of random variables as given independently and identically distributed ${\displaystyle {\mathcal {A}}}$ if the family is given independently and the conditional distributions are all the same. ${\displaystyle {\mathcal {A}}}$ ${\displaystyle P(\{X_{i}\in \cdot \}|{\mathcal {A}})}$
• For example, every family of partial σ-algebras is always given independently , just as every independent family of σ-algebras (in the sense of the independence of a set system ) is always independently given the trivial σ-algebra. ${\displaystyle {\mathcal {A}}}$ ${\displaystyle {\mathcal {A}}}$

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 .