Likelihood function of a hierarchical model

Question

I have the following model:

$$ y\sim\textrm{MvNormal}\left(\mu,\Sigma\right)\\ p=\textrm{logistic}\left(y\right)\\ k\sim\textrm{Binomial}\left(p,n\right) $$

Where $\mu$ and $\Sigma$ are free parameters, and $k$ and $n$ are known. What is the likelihood or log-likelihood function of this model? I know the likelihoods of the multivariate normal and binomial distributions, but how do I combine them to find the likelihood of this hierarchical model?

If there is one, I would like to know the general method for finding likelihoods of these kinds of hierarchical models. Another model I'm interested in is:

$$ y\sim\textrm{MvNormal}\left(\mu,\Sigma\right)\\ \lambda=\textrm{exp}\left(y\right)\\ x\sim\textrm{Poisson}\left(\lambda\right) $$

where $x$ is known.

Answer 1

I came across this question with something similar in mind. I hope my terminology is correct, I don't have a formal stats/math background.

For clarity I will include the index $i = 1, \dots, m$, since I find it confusing to refer to multivariate binomial and Poisson distributions. Hence we can write the first model as $$ k_i \sim \text{Binomial}\left(p_i, n\right) \\ p_i = \text{logistic}(y_i) \\ \mathbf{y} \sim N_m(\mu, \Sigma) $$ while the joint likelihood of the data under this model is the product of the two conditional distributions $$ p(k|y,p,\mu,\Sigma) = p(k|p) \ p(y|\mu, \Sigma) $$ where $p(k|y)$ is the binomial probability mass function, and $p(y|\mu, \Sigma)$ is a multivariate normal density.

Similarly, for $i = 1, \dots, m$, $$ x_i \sim \text{Poisson}(\lambda_i) \\ \lambda_i = \exp(y_i) \\ \mathbf{y} \sim N_m(\mu, \Sigma) $$ the joint likelihood is the product of the conditional distributions $$ p(x|y,\mu,\Sigma) = p(x|y) \ p(y|\mu, \Sigma) $$ where $p(x|y)$ is the Poisson probability mass function, and $p(y|\mu, \Sigma)$ is a multivariate normal density.