Suppose $U\sim N(0,I_p)$, $Y|U\sim N(x(t),\sigma_e^2I_m)$, and the marginal distribution of $Y$ is $f(y)=\int_u f(y|u)f(u)du$.
$x(t)$ is composite function of $U$, basically $x(t)$ is a function of $z(t)$ and $z(t)$ is a function of $U$. I can provide the expression of $x(t)$ in terms of $U$, it's too complicated(involves exponent terms, integrand in dinominator), but I feel it's probably unnecessary. Another info is that the marginal pdf of $y$ has no-closed form expression.
I think what I need to do is generate random variable from $f(y)$, I am reading Monte Carlo method textbook, but there is no detail or probably I am newbie about generating random variable. The textbook is giving an example of normal mixture and the example is given https://stats.stackexchange.com/questions/70855/generating-random-variables-from-a-mixture-of-normal-distributions here too, but it's not similar what I am dealing with.
I have also seen this https://stats.stackexchange.com/questions/508216/how-to-generate-data-from-cdf-which-is-not-in-closed-form?noredirect=1&lq=1 and https://stats.stackexchange.com/questions/466959/multivariate-and-marginal-simulations, but still so confused. The fact is how do I assume the distribution(marginal distribution) of $y$. Or do I need to go for indirect method (eg. accept-reject method)?
Like in the last link they are telling to generate $y_1^{(1)}$ from $f(y_1)$ first. But how do I do that if I don't know the $f(y_1)$
I want to implement the algorithm in r, but I am stuck at the very beginning of making the algorithm.