Metropolis-Hastings

Remarks:

  • 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 of YtY_t leads to repetition of X(t)X^{(t)} at time t+1t+1 (an impossible occurrence in absolutely continuous iid settings)

  • Minimal regularity conditions on both ff and the conditional distribution qq for ff to be the limiting distribution of the chain X(t)X^{(t)}: xsuppfsuppq(yx)supp,f\cup_{x\in \mathrm{supp}\, f}\mathrm{supp}\, q(y\mid x)\supset \mathrm{supp}\,,f.

  • ff is the stationary distribution of the Metropolis chain: it satisfies the detailed balance property.

    K(x,y)=ρ(x,y)q(yx)+(1r(x))δx(y).K(x,y) = \rho(x,y)q(y\mid x)+(1-r(x))\delta_x(y)\,.

The MH Markov chain has, by construction, an invariant probability distribution ff, if it is also an aperiodic Harris chain, then we can apply ergodic theorem to establish a convergence result.

  • A sufficient condition to be aperiodic: allow events such as {X(t+1)=X(t)}\{X^{(t+1)}=X^{(t)}\}.

  • Property of irreducibility: sufficient conditions such as positivity of the conditional density qq.

  • If the MH chain is ff-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.