# Monte carlo integrations with metropolis hastings step

Consider the following problem:

Suppose we want to compute the following integral $$f(y_1|y_2) = \int_{\theta} \int_{x_1} \int_{x_2} f(y_1| x_1,x_2,\theta, y_2) f(x_1,x_2|\theta,y_2) f(\theta | y_2) d x_2 d x_1 d \theta$$

with a monte carlo integration. I am able to sample from $f(\theta | y_2)$ and $f(y_1| x_1,x_2,\theta, y_2)$ but not from $f(x_1,x_2|\theta,y_2)$.

Consider the following sampler: at iterations $t$:

1) sample $\theta^{t}$ from $\theta | y_2$

2) sample $x_1^t$ from $x_1 | x_2^{t-1} , \theta^t ,y_2$ with a Metropolis step

3) sample $x_2^t$ from $x_2 | x_1^{t} , \theta^t ,y_2$ with a Metropolis step

4) sample $y_1^t$ from $y_1| x_1^t,x_2^t,\theta^t, y_2$

Can i consider the set $\{ y_1^t\}_{t=1}^T$ be a set from $f(y_1|y_2)$? (maybe a correlated set)...

It seems strange for me but some simulation that I did shows this is true!!! There is some technical justification, or some paper i can read?