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.
Last modified 3yr ago
Copy link