In this exercise, we will try two MCMC algorithms to sample data following a Wigner semicircle distribution. The probability density function 2 Let's consider a different MCMC algorithm called slice sampler. With slice sampler, we will follow the following steps to draw random samples: Given Xt, the random sample drawn at the t-th step, sample Yt U[O, p(Xt)]. Given Yt above, sample Xt+l uniform distribution on the set {x : 2 Yt}. xt Figure 2: An illustrative example of slice sampler We can show that {Xt : t > 1} is a Markov chain. Can you prove that p(x) is the stationary distribution of this chain? (Hint: similar to the proof of Metropolis-Hastings that we've discussed in class, check detailed balance condition.)