Suppose that I want to generate a proposal $(x^*,y^*,z^*)$ with the following:
$$z^*\sim p(z|\alpha,\beta)$$ $$x^*\sim p(x|z^*,\boldsymbol{\gamma}_x)$$ $$y^*\sim p(y|z^*,\boldsymbol{\gamma}_y),$$ where $\alpha,\beta,\boldsymbol{\gamma}_x,\boldsymbol{\gamma}_y$ are known hyperparameters. Note the dependence of $x^*$ and $y^*$ on $z^*$. How can I appropriately write the acceptance probability for the proposed Metropolis-Hastings sampling scheme?
My attempt:
I know under the single-proposal case, we can write the acceptance probability $\alpha$ for a proposed $\theta$ as
$$\alpha=\min\left\{1,\frac{p(\theta^*|\cdot)p(\theta|\theta^*)}{p(\theta|\cdot)p(\theta^*|\theta)}\right\},$$
where $p(\theta|\cdot)$ denotes the posterior for $\theta$ and $p(\theta|\theta^*)$ denotes the transition probability from $\theta^*$ to $\theta$. This leads me to believe that we can write the joint acceptance probability of the proposed scheme as
$$\alpha=\min\left\{1,\frac{p(z^*,x^*,y^*|\cdot)p(z|z^*)p(x|x^*)p(y|y^*)}{p(z,x,y|\cdot)p(z^*|z)p(x^*|x)p(y^*|y)}\right\}$$
Is there a way to simplify the expression of this acceptance probability further? Thank you.