Ancestral Sampling

When a joint distribution can be factored into a series of conditional probabilities, we can generate samples using ancestral sampling. the basic idea is to generate a sample from the root variable(s) and then sample from the subsequent conditional distributions based on this instantiation.

Consider a joint distribution $Pr(x,y,z)$ over three variables, which (in this particular case) factors as:

$$Pr(x,y,z)=Pr(x)Pr(y|x)Pr(z|y)$$

To sample from this joint distribution, we first draw a sample $x^{\ast }$ from $Pr(x)$. Then we draw a sample $y^{\ast }$ from $Pr(y|x^{\ast })$. Finally, we draw a sample $z^{\ast }$ from $Pr(z|y^{\ast })$.