Basic question about MCMC Metropolis–Hastings algorithm. I am trying to understand the Metropolis–Hastings algorithm and it's connection to Bayesian Analysis. Suppose I want to construct an MCMC MH algrorithm to evaluate my posterior distribution in the my Bayesian Analysis.
I am looking at the step where $\alpha$ is being computed: $$\alpha = \frac{P(\theta^* | \textbf{Y})}{P(\theta^{(i)} | \textbf{Y})}$$ Here I have (for simplicity) assumed that my prposal density in MH algorithm is symmetric.
$P(\theta^* | \textbf{Y})$ is the likelihood of $\theta^*$ given our data. So basically before the MH algorithm we need to assume a distribution for $\theta$, right? And is $P(\theta^* | \textbf{Y})$ our posterior distribution? How do I compute $P(\theta^* | \textbf{Y})$?
Also, feel free to provide an oral (in words) explanation of $P(\theta^* | \textbf{Y})$ and it's connection the posterior distribution we wish to compute. My confusion arises with the fact that we are not able to know our posterior distribution and that is the whole point of MH. So how can comput the likelihood ...