Geoscience Reference
In-Depth Information
(i) Given a current state
x
i
, propose a move to
x
with probability
q.
x
i
;
x
/:
(3.4)
(ii) Now, compute forward observations
y
by running the forward model on the
proposed state
x
.
(iii) Compute the
Hastings ratio
/
D
P.
y
j
x
/P.
x
/q.
x
;
x
i
/
.
x
i
;
x
/
;
(3.5)
P.
y
i
j
x
i
/P.
x
i
/q.
x
i
;
x
(iv) And accept the proposed move to
x
with probability
Q.
x
i
;
x
/
D min
.1;.
x
i
;
x
//:
(3.6)
Note that if the transition probability is symmetric and stationary, then the probabil-
ity
q.
x
i
;
x
/
is identical to
q.
x
;
x
i
/
and the Hastings ratio simplifies to the Metropolis
update
/
D
P.
y
j
x
/P.
x
/
.
x
i
;
x
(3.7)
P.
y
i
j
x
i
/P.
x
i
/
Note the similarity between the numerator and denominator of (
3.7
)andthe
numerator in (
3.3
). It is precisely the accept/reject criterion that allows MCMC to
sample from the un-normalized posterior
.
It is common to construct proposals of the form
x
D
x
i
C
P.
x
j
y
/
where
N.0;
†
x
/
.
Here,
†
x
is the variance of the proposal distribution-specification of this is one
of the subtleties involved in constructing an MCMC algorithm. The choice of
transition probability
may assume any
form, consistent with the characteristics of the system of study. The requirement
for any MCMC algorithm is that the Markov chain with (Metropolis or Hastings)
updates converges (in the limit of infinite number of proposal steps) to sampling
the stationary invariant posterior distribution (ergodicity). Ergodicity is guaranteed
if the MCMC algorithm is properly constructed. In practice, convergence is not
assured if the form and properties of the posterior distribution are unknown (black
box MCMC). The construction of an MCMC algorithm involves a number of
decisions as to the form and characteristics of the transition probability distribution,
as well as the specifics of the Metropolis-Hastings update. We discuss such practical
issues in the next section.
q.
x
;
/
is flexible and the likelihood
P.
y
j
x
/
3.3
Practical Issues
The only fundamental requirement for proper MCMC simulation is that the user
construct a chain that is Markov with transition probabilities and updates that ensure
ergodicity to the invariant posterior (target) distribution. The practical usability of an
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