Biomedical Engineering Reference
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where L(a; ; t
0
) is the natural logarithm of likelihood function 14:
L(a; ; t
0
) = ln(p(sja; ; M
k
; t
0
)):
(18)
Solving equations 17 gives the maximum likelihood estimations of pa-
rameters (a
;
) for the window with a xed t
0
and data s. If the data
is informative, then the likelihood would sharply peak around the point
(a
;
) in the parameter space and die o quickly as we move away from
(a
;
). It is natural to Taylor expand 18 around (a
;
):
L(a; ; t
0
)
L(a
;
; t
0
)
+
2
h
i
@
2
L
@a
2
a
;
(aa
)
2
+
@
2
L
@
2
a
;
(
)
2
j
j
(19)
+
@
2
L
@a@
a
;
(aa
)(
)
= L(a
;
; t
0
) +
2
(XX
)
0
rrL(a
;
; t
0
)(XX
);
j
a
where X =
, and:
2
4
@
2
L
3
@
2
L
@a@
@a
2
5
a
;
a
;
rrL(a
;
; t
0
) =
(20)
@
2
L
@a@
@
2
L
@
2
a
;
a
;
is the Hessian matrix evaluated at (a
;
).
It follows from 19 that the leading term of the likelihood function in 14
is approximately:
p(sja; ; M
k
; t
0
)
= exp [L(a; ; t
0
)]
exp
(21)
L(a
;
; t
0
) +
2
(XX
)
0
rrL(a
;
; t
0
)(XX
)
= e
L(a
;
;t
0
)
e
2
(XX
)
0
rrL(a
;
;t
0
)(XX
)
:
This implies that the likelihood function looks like a multivariate normal
distribution in parameter space, centered at (a
;
) with the following
uncertainties, if a and are not coupled:
@
2
L
@a
2
1=2
a
=
j
a
;
(22)
1=2
@
2
L
@
2
=
j
:
a
;
Moreover, the approximation in equation 21 provides a possibly easy
way to compute the evidence in equation 15 in the sense that if the prior
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