Biomedical Engineering Reference
In-Depth Information
Thus, any expectation of a function
f
with respect to the posterior can
be estimated from a delayed series of the realization of
C
1
:
f, E
[
f
]=
V
∀
p
(
q
|
b
obs
)
·
f
(
q
)
·
d
q
(3.88a)
f
1
,
i
=
N
1
(
i
)
q
(3.88b)
N
−
M
i
=
M
+1
where the first
M
iterations of the Markov chain represent the burn-in period
(i.e., the convergence process of
C
1
from random initial values to the posterior
distribution).
Other chains
C
i
,for
i>
1, are called auxiliary chains: they prevent the
principal chain from getting trapped in a local mode. Each chain
C
i
is asso-
ciated with a temperature parameter
T
i
, such that
T
1
=1and
T
i
<T
i
+1
.An
unnormalized probability distribution
π
i
is then defined for each chain
C
i
,so
that “hot” auxiliary chains are associated with probability distributions that
become easier and easier to sample:
q
|
b
obs
)
1
/T
i
.
π
i
=
p
(
(3.89)
Transition probabilities
p
(
j
)
i
are defined according to the Metropolis samp-
ler [52], so that for each chain
C
i
, the distribution of its elements will converge
to the probability distribution
π
i
. A proposal value
q
i
is drawn at random
(
j
)
i
around
and is accepted with probability
r
:
r
=
min
1
,
q
.
q
i
π
i
(
)
(3.90)
(
j
)
i
π
i
(
q
)
If the proposal value is accepted, it becomes the new realization of the
chain; otherwise, the previous value is retained and the chain does not move.
Movement between the chains is allowed by PT moves. Two contiguous
chains
C
i
and
C
i
+1
are chosen at random and their realizations are swapped
with probability
r
=
min
1
,
π
i
(
.
(
j
)
i
+1
)
(
j
)
i
q
·
π
i
+1
(
q
)
(3.91)
(
j
)
i
(
j
)
i
+1
)
π
i
(
q
)
· π
i
+1
(
q
For a variable number of dipoles, we added reversible jump (RJ) moves
[53] to the previous sampler. The parameters to be estimated now take the
form (
k,
q
k
), where
k
is the unknown number of dipoles, and
q
k
is the set of
these
k
dipoles.Themovefrom(
i,
q
i
) to the proposal value (
j,
q
j
) is realized
by drawing a random vector
u
i
independently of
q
i
, and setting
q
j
to some
deterministic function
f
i
of
q
i
and
u
i
. This move and the reverse move must
satisfy
dim
q
i
+
dim
u
i
=
dim
q
j
+
dim
u
j
.
(3.92)
