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
$Y_t$
$X^{(t)}$
at time
$t+1$
(an impossible occurrence in absolutely continuous iid settings)
• Minimal regularity conditions on both
$f$
and the conditional distribution
$q$
for
$f$
to be the limiting distribution of the chain
$X^{(t)}$
:
$\cup_{x\in \mathrm{supp}\, f}\mathrm{supp}\, q(y\mid x)\supset \mathrm{supp}\,,f$
.
• $f$
is the stationary distribution of the Metropolis chain: it satisfies the detailed balance property.
$K(x,y) = \rho(x,y)q(y\mid x)+(1-r(x))\delta_x(y)\,.$
The MH Markov chain has, by construction, an invariant probability distribution
$f$
, 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
$\{X^{(t+1)}=X^{(t)}\}$
.
• Property of irreducibility: sufficient conditions such as positivity of the conditional density
$q$
.
• If the MH chain is
$f$
-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.