Biomedical Engineering Reference
In-Depth Information
probability of exactly
J
events in
(
0
,
T
]
may be computed as
J
'
j
J
'
j
p
(
N
0:
T
)
1
k
j
=
p
(
u
j
∈
(
t
k
j
−
1
,
t
k
j
]
,
j
=
1
, ··· ,
J
∩
N
(
T
)=
J
)
1
k
j
=
=
k
=
Pr
(
∩
1
E
k
)
=
'
k
k
−
1
=
Pr
(
E
k
|∩
j
=
1
E
j
)
Pr
(
E
1
)
=
2
'
k
dN
(
t
k
)
1
−
dN
(
t
k
)
=
1
[
l
(
t
k
|
H
k
)
k
]
[
1
−
l
(
t
k
|
H
k
)
k
]
+
o
(
∗
)
=
J
'
j
=
dN
(
t
k
j
)
'
1
−
dN
(
t
l
)
=
1
[
l
(
t
k
j
|
H
k
j
)
k
j
]
l
=
k
j
[
1
−
l
(
t
l
|
H
l
)
l
]
+
o
(
∗
)
J
'
j
dN
(
t
k
j
)
'
l
=
1
[
l
(
t
k
j
|
H
k
j
)
k
j
]
exp
{−
l
(
t
l
|
H
l
)
l
}
+
o
(
∗
)
=
=
k
j
exp
J
Â
j
Â
l
=
k
j
=
log l
(
t
k
j
|
H
k
j
)
dN
(
t
k
j
)
−
l
(
t
l
|
H
l
)
l
=
1
exp
J
Â
j
·
log
k
j
+
o
(
∗
)
=
1
(9.11)
where, because the
k
are small, we have used the approximation
[
1
−
l
(
k
)
k
]
≈
exp
{−
l
(
k
)
k
}
and
∗
=
max
k
k
. It follows that the probability density of exactly
these
J
spikes in
(
0
,
T
]
is
exp
J
Â
j
⎡
⎣
Â
l
=
k
j
log l
(
t
k
j
|
H
k
j
)
dN
(
t
k
j
)
−
l
(
t
l
|
H
l
)
l
=
1
p
(
N
0:
T
)=
lim
∗
→
J
'
j
0
1
j
=
⎤
exp
J
Â
j
⎦
o
(
∗
)
J
'
j
·
log
k
j
+
=
1
1
j
=
exp
T
0
du
T
Q.E.D.
(9.12)
Proposition 1 shows that the joint probability density of a spike train process can
be written in a canonical form in terms of the conditional intensity function [3, 8,
11]. That is, when formulated in terms of the conditional intensity function, all
point process likelihoods have the form given in Equation (9.8). The approximate
probability density expressed in terms of the conditional intensity function (Equation
(9.11d)) was given in [5]. The proof of Proposition 1 follows the derivation in [1].
The insight provided by this proof is that correct discretization for computing the
=
log l
(
u
|
H
u
)
dN
(
u
)
−
l
(
u
|
H
u
)
.
0
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