Biomedical Engineering Reference
In-Depth Information
If the noise is Poisson type, then H
1
implies that the local rate r
k
is:
r
k
= ax
k
+ r
0
;
(9)
where r
0
characterizes the dark current.
We wish to see that, given the data s, whether H
1
is more favorable or
H
0
is. This can be done by computing the odds, i.e., taking the ratio of
two probabilities p(H
1
js) and p(H
0
js):
p(H
1
p(H
0
js; t
0
)
=
p(M
1
js; t
0
)
js; t
0
)
p(M
0
js; t
0
)
:
(10)
If the odds is large compared to one, we can be condent that there is
a peak in the window, while if it is approximately one we interpret that
as saying there is only weak evidence of a peak in the window (because
a = 0 is a possible estimate of the peak intensity, which we interpret as `no
peak'). Therefore, one may set a threshold for peak detection.
In order to compute the odds in 10, let us invoke Bayes' theorem:
p(XjY; I) =
p(YjX; I)p(XjI)
p(YjI)
:
(11)
Identify Y as data, s, observed in the window, and X as the model M
k
:
js; t
0
) =
p(sjM
k
; t
0
)p(M
k
jt
0
)
p(M
k
:
(12)
p(sjt
0
)
If there is no reason to prefer M
0
over M
1
, then one should assign equal
prior probabilities, i.e., p(M
0
jt
0
) = p(M
1
jt
0
) = 1=2. Then, when take the
ratio in 10, the denominator would cancel and we have the simple result
that:
p(H
1
js; t
0
)
p(H
0
js; t
0
)
=
p(M
1
js; t
0
)
js; t
0
)
=
p(sjM
1
; t
0
)
p(sjM
0
; t
0
)
:
(13)
p(M
0
Hence, we need to calculate the probability of observing the data s
given the model M
k
, p(sjM
k
; t
0
), a quantity called marginal likelihood,
or evidence, and is not conditional on any parameters. It can be computed
through marginalization.
Since, M
k
implies a particular noise process, we may characterize it by
parameter (e.g., the variance and mean for a Gaussian process or the
`dark' current rate r
0
for a Poisson process), and notice that ion counts at
dierent times are independent, then the likelihood function, i.e., the
probability of observing the particular count sequence s = (s
1
; s
2
s
N
)
given model M
k
and its associated parameters (a; ) is simply:
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