Let's consider a toy example of applying a Markov Chain Monte Carlo (MCMC) method. Let $X$ be a discrete random variable whose unnormalized probability mass function is $$ \tilde{p}(X=1)=4, \tilde{p}(X=2)=2, \tilde{p}(X=3)=3, \tilde{p}(X=4)=3 $$ Our goal is to evaluate $\mathbf{E}[f(X)]$ where $f(x)=x^{2}-3 .$ To apply the Metropolis-Hasting algorithm, define two proposal distributions $q(i, j)\left(=q_{i j}\right)$ and $\tilde{q}(i, j)\left(=\tilde{q}_{i j}\right)$ where $$ q(i, i+1)=q(i, i-1)=\frac{1}{2}, \quad \tilde{q}(i, i+1)=\frac{2}{3}, \tilde{q}(i, i-1)=\frac{1}{3} . $$ (If $i+1=5$, consider 5 as 1. Also, if $i-1=0$, consider 0 as 4.)
Using the Metropolis-Hasting algorithm, how can one construct Markov chains for proposal distributions $q$ and $\tilde{q}$ when we set the initial state $X_{0}=1$?
It seems that in order to solve the question, transition matrices are important for Markov chains, but how one can come up with that? Are there any solid ideas to evaluate $\mathbf{E}[f(X)]$ via Metropolis-Hasting approach?