Biomedical Engineering Reference
In-Depth Information
balance out the one-dimension increase of the proposed new model. When the
proposal jump time
is near the boundaries, we use a constant extrapolation
for '(t) in Equation (7.15). That is, we set '(
0
) = '(
1
) when
is before
the first jump time
1
, and '(
J+1
) = '(
J
) when
is between the last two
jump times
J1
and
J
.
Let v =
'(
1
);'(
2
);:::;'(
J
)
and v
=
'(
1
);'(
2
);:::;'(
J+1
)
be
the current and proposal vector of coecient values, with dimension J and
J + 1, respectively. The acceptance ratio is given by
v
jnfv
g;w
2
;D
aug
v jnfvg;w
2
;D
aug
(u)
;
@v
@(v;u)
R
b
=
where ( j nfg;w
2
;D
aug
) is the posterior distribution; (u) = 1f
0
<
u <
0
g=(2
0
) is the density function of uniform distribution; and the last
term is the Jacobian of the one-to-one transformation from (v;u) to v
. The
acceptance probability is minf1;R
b
g.
7.3.2.3
Death Move
Choose an index j uniformly from the current jump point set f1; 2;:::;J1g
and remove the jump time
j
. The new J 1 jump times are relabeled as
follows: jump times before
j
keep their indices; jump times after
j
decrease
their indices by 1. Denote the new jump times as
j
, j = 1; 2;:::;J1. After
the deletion of a jump time, the two constant pieces on interval [
j1
;
j+1
)
are combined into one. The proposal of the coecient value is the inverse of
the birth move proposal. Given values of '(
j
) and '(
j+1
), solve unknown
'(
j
) and u from the following equations:
'(
j
) =
1
'(
j1
) +
2
'(
j
) + u
;
'(
j+1
) =
1
'(
j
) u
+
2
'(
j+1
);
yielding
n
1
o
1
2
1
2
'(
j
) +
1
'(
j+1
)
2
1
'(
j
) =
2
'(
j1
) +
1
'(
j+2
)
; (7.16)
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