Biomedical Engineering Reference
In-Depth Information
local probability of a spike event is given by the conditional intensity function. An
alternative derivation of Equation (9.8) can be obtained directly using Equation (9.7)
[3].
If the probability density in Equation (9.8) depends on an unknown
q
-dimensional
parameter
q to be estimated then, Equation (9.8) viewed as a function of
q given
N
0:
T
is the likelihood function defined as
L
(
q
|
N
0:
T
)=
p
(
N
0:
T
|
q
)
exp
T
0
du
T
(9.13)
=
log l
(
u
)
|
H
u
,
q
)
dN
(
u
)
−
l
(
u
|
H
u
)
.
0
The logarithm of Equation 9.13 is the log likelihood function defined as
T
log
L
(
q
|
N
0:
T
)=
l
u
(
q
)
du
(9.14)
0
where
l
t
(
q
)
is the integrand in Equation (9.14) or the
instantaneous
log likelihood
defined as
dN
)
dt
−
(
t
l
t
(
q
)=
log
[
l
(
t
|
H
t
,
q
)]
l
(
t
|
H
t
,
q
)
.
(9.15)
Given a model for the spike train, defined either in terms of the conditional intensity
function or the interspike interval probability density, the likelihood is an objective
quantity that offers a measure of
rational belief
[9, 31]. Specifically, the likelihood
function measures the relative preference for the values of the parameter given the
observed data
N
0:
T
. Similarly, the instantaneous log likelihood in Equation (9.15)
may be viewed as measuring the instantaneous accrual of
information
from the spike
train about the parameter q. We will illustrate in the applications in Section 9.3 how
methods to analyze neural spike train data may be developed using the likelihood
function. In particular, we will use the instantaneous log likelihood as the criterion
function in the point process adaptive filter algorithm presented in Section 9.3.3.
9.2.3
Summarizing the likelihood function: maximum likelihood esti-
mation and Fisher information
If the likelihood is a one or two-dimensional function it can be plotted and com-
pletely analyzed for its information content about the model parameter q.When
the dimension of q is greater than 2, a complete analysis of the likelihood func-
tion by graphical methods is not possible. Therefore, it is necessary to summarize
this function. The most common way to summarize the likelihood is to compute
the maximum likelihood estimate of the parameter q. That is, we find the value of
this parameter that is most likely given the data. This corresponds to the value of q
that makes Equation (9.13) or equivalently, Equation (9.14) as large as possible. We
define the maximum likelihood estimate q as
q
=
arg max
q
L
(
q
|
N
0:
T
)=
arg max
q
log
L
(
q
|
N
0:
T
)
.
(9.16)
With the exception of certain elementary models the value of qthat maximizes Equa-
tion (9.16) has to be computed numerically. In most multidimensional problems it is
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