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Suppose $X$ and $Y$ are two discrete random variables. The law of total probability states that: $$ p(x) = \sum\limits_y {p(x,y) = } \sum\limits_y {p(x|y)p(y)} $$

Now suppose we have another random variable $Z$ and we want to do marginalization on both $Y, Z$. My question is, what would be the statement of the law of total probability when we have more extra variables?

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The next iteration of the rule would be:

$$p(x) = \sum_y \sum_z p(x,y,z) = \sum_y \sum_z p(x|y,z) p(y|z) p(z).$$

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  • $\begingroup$ Thank you for your answer. Is there any numerical example of how this iterated expression can be computed? $\endgroup$
    – sci9
    Oct 6, 2020 at 8:30

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