Editing Bayesian network (section)

===Factorization definition===
''X'' is a Bayesian network with respect to ''G'' if its joint [[probability density function]] (with respect to a [[product measure]]) can be written as a product of the individual density functions, conditional on their parent variables:{{sfn|Russell|Norvig|2003|p=496}}

: <math> p (x) = \prod_{v \in V} p \left(x_v \,\big|\,  x_{\operatorname{pa}(v)} \right) </math>

where pa(''v'') is the set of parents of ''v'' (i.e. those vertices pointing directly to ''v'' via a single edge).

For any set of random variables, the probability of any member of a [[joint distribution]] can be calculated from conditional probabilities using the [[chain rule (probability)|chain rule]] (given a [[topological ordering]] of ''X'') as follows:{{sfn|Russell|Norvig|2003|p=496}}

: <math>\operatorname P(X_1=x_1, \ldots, X_n=x_n) = \prod_{v=1}^n \operatorname P \left(X_v=x_v \mid X_{v+1}=x_{v+1}, \ldots, X_n=x_n \right)</math>

Using the definition above, this can be written as:

: <math>\operatorname P(X_1=x_1, \ldots, X_n=x_n) = \prod_{v=1}^n \operatorname P (X_v=x_v \mid X_j=x_j \text{ for each } X_j\, \text{ that is a parent of } X_v\, )</math>

The difference between the two expressions is the [[conditional independence]] of the variables from any of their non-descendants, given the values of their parent variables.