Bayesian Networks

Resources:

• https://www.youtube.com/watch?v=TuGDMj43ehw
• page=225

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The conditional probability in a Bayesian network is given by its parents

$$P(X|\text{Parents}(X))$$

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Inference in a Bayesian Network

Doing it naively means calculating the same values multiple times, leading to a runtime of $O(2^n)$.

Variable Elimination

We eliminate variables that are neither:

1. Part of the query (what you’re trying to find out)
2. Evidence variables (what you already know or observe)

The eliminated variables are often called hidden variables, and eliminating them simplifies the computation.

Example: We want to find $P(Y)$ given

$$P(W,X,Y,Z)=P(W)P(X|W)P(Y|X)P(Z|Y)$$

We sum out all possible variations of everything except $Y$

$$P(Y)=\sum _w\sum _x\sum _{z}P(w)P(x|w)P(Y|x)P(z|Y)$$

By setting $f_w(x)={\sum }_{w}P(w)P(x|w)$, and $f_x(Y)={\sum }_x{f}_w(x)P(Y|x)$ we get

$$P(Y)=\sum _z{f}_x(Y)P(z|Y)$$

Which can be visualized as a Bayesian network like this: