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each a goodness-of-fit analysis. We believe the goodness-of-fit assessment is a cru-
cial, yet often overlooked, step in neuroscience data analyses. This assessment is
essential for establishing what data features a model does and does not describe.
Perhaps, most importantly, the goodness-of-fit analysis helps us understand at what
point we may use the model to make an inference about the neural system being stud-
ied and how reliable that inference may be. We believe that greater use of the like-
lihood based approaches and goodness-of-fit measures can help improve the quality
of neuroscience data analysis. Although we have focused here on analyses of single
neural spike train time-series, the methods can be extended to analyses of multiple
simultaneously recorded neural spike trains. These latter methods are immediately
relevant as simultaneously recording multiple neural spike trains is now a common
practice in many neurophysiological experiments.
9.5
Appendix
Lemma 1.
Given
n
events
E
1
,
E
2
, ··· ,
E
n
in a probability space, then
Pr
E
i
|∩
1
E
j
Pr
n
'
i
n
i
=
i
−
1
Pr
(
∩
1
E
i
)=
(
E
1
)
.
(9.42)
j
=
=
2
Proof
: By the definition of conditional probability for
n
=
2,
Pr
(
E
1
∩
E
2
)=
Pr
(
E
2
|
E
1
)
Pr
(
E
1
)
. By induction
Pr
E
i
|∩
1
E
j
Pr
n
−
Pr
∩
1
E
i
=
'
i
n
−
1
i
−
1
(
E
1
)
.
(9.43)
i
=
j
=
=
2
Then
Pr
E
n
|∩
E
i
Pr
∩
i
=
1
E
i
i
n
−
1
i
=
1
n
−
1
Pr
(
∩
1
E
i
)=
=
Pr
E
i
1
E
j
Pr
Pr
E
n
1
E
i
n
'
i
−
n
−
1
i
i
−
1
j
=
|∩
|∩
(
E
1
)
=
=
(9.44)
=
1
Pr
E
i
1
E
j
Pr
n
'
i
i
−
1
j
=
|∩
(
E
1
)
.
Q
.
E
.
D
.
=
=
1
Acknowledgments.
Support for this work was provided in part by NIMH grants
MH59733, MH61637, NIDA grant DA015664 and NSF grant IBN-0081458. We
are grateful to Satish Iyengar for supplying the retinal spike train data analyzed in
Section 9.3.1.
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