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Let $f(x,y,z)$ be the joint density function. I found a reference that this joint density can be written as

$f(x,y,z) = f_1(x|y,z)f_2(y|z)f_3(z)$

I'm wondering if there are alternative forms to decompose $f(x,y,z)$. Any results would be much appreciated.

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    $\begingroup$ Any permutation of $x$, $y$, and $z$ in the density decomposition is also corect $\endgroup$
    – Xi'an
    Commented Jun 14, 2021 at 10:29
  • $\begingroup$ If $x,y,z$ are causally related, sometimes this factorization can be truncated and written more compactly. If they are not causally related, then this expression or, as Xi'an noted, any permutation of this expression, is your option. See "Causal Inference in Statistcs: A Primer", pp. 29-30. $\endgroup$
    – Adrian Keister
    Commented Jun 14, 2021 at 13:07
  • $\begingroup$ Many thanks for your replies! $\endgroup$
    – user0131
    Commented Jun 15, 2021 at 0:51
  • $\begingroup$ Another question. Is $f(x,y|z)=f_1(x|y,z)f_2(y|z)=f_3(y|x,z)f_4(x|z)$ correct? I'm not confident about conditioning part... $\endgroup$
    – user0131
    Commented Jun 15, 2021 at 6:09

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