 Solution

Write the target posterior density as $\pi (\theta )$ and the proposal distribution density as $q({\theta }^{\prime }\mid \theta )$. The Metropolis–Hastings algorithm accepts a proposed move withFor distinct states,which is symmetric in $\theta$ and ${\theta }^{\prime }$. The rejection probability supplies the diagonal part, so the entire transition kernel satisfies detailed balance. Integrating the detailed-balance identity over the starting state proves $\int \pi (\theta )P(\theta ,d{\theta }^{\prime })=\pi ({\theta }^{\prime })d{\theta }^{\prime }$. Hence the posterior is a stationary distribution; an irreducible Markov chain that is also an aperiodic Markov chain converges uniquely to it.