# Conditional Independence

## Definition

Conditional Independence describes a general condition for three or more random variables, where the distributions of a subset of variables conditional on the knowledge of the complement are independent. The principle finds application is many areas of [[Risk Model] development as it simplifies the Dependency Structure

Conditional Independence is an important consideration in the context of the development of Probabilistic Graphical Models

In the context of a Credit Network model is a stochastic framework where the realisation of risk factors and associated risk events for individual entities of the network is independent once we condition on the realization of common (typically, but not necessarily macro) factors.

## Notation

In this class of model the rating state of each individual entity is completely determined by an implied credit quality indicator

${\displaystyle R_{k}^{i}=G(W_{k}^{i};A_{k}^{mn})}$
${\displaystyle W_{k}^{i}=F(Z_{k},U_{k},\epsilon _{k}^{i})}$

## More General Models

In more complicated credit contagion models, credit risk cascades are precipitated directly by default events