I am reading about the MCMC but now I got a lot of questions.
Firstly, It says we could construct a markov chain which satisfy the detailed balance:p(z)T(z,z')=p(z')T(z',z) and we could sample from the proposal distribution T after we converge to the stationary distribution p(z). But why shouldn't we sample from p(z) but from the T(z,z')?
Secondly, why after we constructed this markov chain, it will ensure converge to the target distribution we want but not some other stationary distribution? The book says it satisfy the aperiodic and ergodicity. How the two properties make sure the stationary distribution is the distribution we want?
I am getting stuck here for several days. Thank you for somebody helping me out.


1 Answer 1


First - of course we should sample from p(z), but in some cases it is impossible and that is where MCMC is applicable.

Second - the correct stationary distribution is ensured by detailed balance condition. If further conditions are met (in simple case of finite state space it is enough that state is aperiodic and irreducible) the stationary distribution is unique, moreover regardless of the starting distribution there will be convergence to stationary distribution. It is quite nicely explained in Wikipedia https://en.wikipedia.org/wiki/Markov_chain#Time-homogeneous_Markov_chain_with_a_finite_state_space


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