I was wondering if someone could explain the following passage from my textbook.
In the third paragraph from the bottom it is stated that the simulated values come from the marginal distributions. How is this so? We simulate the value x2 from the condition density f(x2|x1)..
Also, there is mention of independence and again because of the dependence of the value of x2 on x1 and so forth, I do not see how this occurs.
Would greatly appreciate your help
This cascading simulation method is the traditional approach to simulating a vector. Simulate one component from its marginal, then a second component given the first one, a third one given the first two ones and so on...
This is a correct method in the sense that the vector is now generated from $$f(x_1)f(x_2|x_1)\cdots f(x_n|x_1,\ldots,x_{n-1})=f(x_1,\ldots,x_n)$$ (with a terribly confusing abuse of notation in resorting to the generic $f$ for all densities).
The rhs of the equation is the joint density of the vector, which can be decomposed in the product on the lhs but also on any other product of the form $$f(x_1,\ldots,x_n)=f(x_{i_1})f(x_{i_2}|x_{i_1})\cdots f(x_{i_n}|x_{i_1},\ldots,x_{i_{n-1}})$$ where $(i_1,\ldots,i_n)$ is any permutation of $(1,2,\ldots,n)$. This means in particular that the marginal distribution of any component of the vector is associated with the same joint $$f(x_{i_1})=\int f(x)\,\text{d}x_{-i_1}$$ and thus that simulating from the joint returns simulation from any marginal when considering only the corresponding component from the vector. It is thus correct to state that, while the simulation ran by first simulating $x_1$ from $f(x_1)$ and then $x_2$ given $x_1$ as from $f(x_2|x_1)$, $x_2$ is also a simulation from $f(x_2)$. Simply because $$f(x_2) = \int f(x_2|x_1)f(x_1)\,\text{d}x_1$$
