I am now studying the Metropolis-Hastings algorithm and I want to apply it in order to made a Bayesian Inference of a function $y=f(x)$ to a dataset $D=\{x_i,y_i\}$. Five parameters of the function need to be determined $\theta=\{\theta_1,...,\theta_5\}$.
Here I want to first introduce how to develop the M-H algorithm.
I start from an initial guess of the model parameters $\theta^{(0)}$ and, using a candidate distribution (in my case a Multivariate Normal Distribution centred in $\theta^{(0)}$ with a certain variance $\delta$ that has to be tuned), select a candidate $\theta^{(*)}$. Since the candidate distribution is symmetrical, the acceptance probability is the ratio:
\begin{equation} \alpha=min\{1, \frac{\pi(\theta^{(*)})}{\pi(\theta^{(0)})} \} \end{equation} Where $\pi(\bullet)$ is the target function, proportional to the Posterior Distribution.
$\theta^{(*)}$ is accepted ($\theta^{(1)}=\theta^{(*)}$) if $\alpha$ is bigger than $u=U(0,1)$, otherwise it is rejected ($\theta^{(1)}=\theta^{(0)}$). A new candidate can be drawn and the process repeated.
In case I want to use the M-H algorithm in a inference problem, the target function $\pi(\theta)$ can be substituted by the Likelihood function $L(\theta)$.
However, if I want to made a Bayesian inference, it is not really clear to me the role of the prior distributions associated to the elements $\theta_i$ of the vector parameter. How do they figure in the acceptance probability?
Thanks for the answers and the comments