Goal of MCMC

Given a pmf $\pi$ on some state space $S$, how do we draw a sample form $\pi$ or compute the expected value of $\pi$ (i.e. $X\sim\pi$ how do we compute $E_{\pi}[X]$)? Well we know that:

• $\pi$ is known up to a multiplicative constant (this is based on the assumptions made in Bayesian Statistics)
• Bayesian Inference is one application The Metropolis Hastings Algorithm, helps us achieve finding $\pi$.