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I am reading Mathematics for Machine Learning and, in the Summary Statistics and Independence section, the author derives the conditional independence of two random variables given a third RV using the product rule of probability $ p(\boldsymbol{x},\boldsymbol{y}) = p(\boldsymbol{y}\vert\boldsymbol{x})p(\boldsymbol{x})$ (Eq 6.22) as follow

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Now, I don't understand how the author derives Equation 6.56 from the product rule of probability (Eq 6.22).

Can someone please show me the full derivation and explain the intuition (if any) ?

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Using conditional probability formulas: $$\begin{align}p(x,y|z)=\frac{p(x,y,z)}{p(z)}=\frac{p(x|y,z)p(y,z)}{p(z)}=\frac{p(x|y,z)p(y|z)p(z)}{p(z)}=p(x|y,z)p(y|z)\end{align}$$

Intuitively, the product rule still holds when we add any number of RVs to the given sides of each product term in both LHS and RHS.

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  • $\begingroup$ Thank you, makes sense! $\endgroup$ – pp492 Nov 17 '19 at 18:36

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