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Math and science::INF ML AI

Belief networks

A belief network (called a Bayesian network by Koller) is a grapical model that represents a distribution of the form
\[ p(x_1, ..., x_D) = \prod_{i=1}^{D} p(x_i \vert pa(x_i)) \\\text{where } pa(x_i) \text{ represents the parental variables of variable } x_i\]

Parental variables is just a term for the variables that appear as conditional variables for a given variable.

A belief network represents a distribution as a directed graph where the \( i \)th node corresponds to the factor \( p(x_i \vert pa(x_i)) \), with directed connections from all \( pa(x_i) \) nodes to the \( i \)th node. 

According to the rules above, a distribution \( p(x_1, x_2, x_3, x_4) \), one with no independence statements, can be modelled with a cascade graph like so:


Without loss of generality, using Bayes' rule the distribution \( p(x_1, ..., x_n) \) can be represented as:

p(x_1, ..., x_n) &= p(x_1 \vert x_2, ..., x_n)p(x_2, ..., x_n) \\
&= p(x_1 \vert x_2, ..., x_n)p(x_2 \vert x_3, ..., x_n)p(x_3, ..., x_n) \\
&= p(x_n) \prod_{i=1}^{n-1} p(x_i \vert x_{i+1}, ..., x_n)

More accurately, a belief network represents a set of distributions that have the same dependency relationships, not just a single distribution with speficied probabilities.


Deleting any connection represents the existance of some independence. For example:


Bayesian Reasoning and Machine Learning
David Barber