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_tYt​
 leads to repetition of X(t)X^{(t)}X(t)
 at time t+1t+1t+1
 (an impossible occurrence in absolutely continuous iid settings)
Minimal regularity conditions on both fff
 and the conditional distribution qqq
 for fff
 to be the limiting distribution of the chain X(t)X^{(t)}X(t)
: ∪x∈supp fsupp q(y∣x)⊃supp ,f\cup_{x\in \mathrm{supp}\, f}\mathrm{supp}\, q(y\mid x)\supset \mathrm{supp}\,,f∪x∈suppf​suppq(y∣x)⊃supp,f
.
​fff
 is the stationary distribution of the Metropolis chain: it satisfies the detailed balance property.
K(x,y)=ρ(x,y)q(y∣x)+(1−r(x))δx(y) .K(x,y) = \rho(x,y)q(y\mid x)+(1-r(x))\delta_x(y)\,.K(x,y)=ρ(x,y)q(y∣x)+(1−r(x))δx​(y).
The MH Markov chain has, by construction, an invariant probability distribution fff
, 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)}\}{X(t+1)=X(t)}
.
Property of irreducibility: sufficient conditions such as positivity of the conditional density qqq
.
If the MH chain is fff
-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.
MCMC diagnostics
Metropolis
Last modified 4yr ago