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I have seen this approach link in different places. Hence, I suppose that the following method is correct.

Let $f_{X,Y,Z}(x,y,z)=f_{X|Y,Z}(x|y,z)f_Y(y)f_Z(z)$ and I can easily sample from these 3 densities.

To sample from the marginal $f_X(x)$ we can do:

  1. For $j=1,\cdots,N$
  2. Sample $(y,z)^j=y^j \sim f_Y, z^j \sim f_Z$
  3. Sample $(x,y,z)^j = x^j \sim f_{X|Y,Z}(\cdot|y^j,z^j)$
  4. Keep only $\{x\}$

Is it correct?

My question is, can this method be used to sample from $f_{X|Y}(x|y)$ for example?

user1571823
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    That is correct! – jbowman Aug 12 '21 at 18:54
  • what about f(x|y)? because i guess keeping {x,y} is equivalent to f(x,y). Right? – user1571823 Aug 12 '21 at 19:08
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    By definition, the marginal density is what you obtain when you ignore the other variables. Thus, if your algorithm just happens to generate one or more of those other variables, you are under no obligation to pay attention to them. – whuber Aug 12 '21 at 19:20

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