- A sample produced by the above algorithm differs from an iid sample. For one thing, such a sample may involve repeated occurrences of the same value, since rejection ofleads to repetition ofat time(an impossible occurrence in absolutely continuous iid settings)
- Minimal regularity conditions on bothand the conditional distributionforto be the limiting distribution of the chain:.
- is the stationary distribution of the Metropolis chain: it satisfies the detailed balance property.
- A sufficient condition to be aperiodic: allow events such as.
- Property of irreducibility: sufficient conditions such as positivity of the conditional density.
- If the MH chain is-irreducible, it is Harris recurrent.
- A somewhat less restrictive condition for irreducibility and aperiodicity.
In the following sections, let's introduce other versions of Metropolis-Hastings.