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
Yt
leads to repetition of
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)
:
∪x∈suppfsuppq(y∣x)⊃supp,f
.
f
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).
The MH Markov chain has, by construction, an invariant probability distribution