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model based on the difference in probability of making a given decision is discussed
later.
The simplest form of the stochastic mathematical model in a decision-making con-
text has individual accumulated data to support the two hypotheses and is given by
( 3.56 ). The integrated value of the evidence at time t is denoted by the dynamic variable
Q
(
)
(
) =
.
t
and we assume that the initial state is unbiased so that Q
0
0
The DDM with
unbiased evidence is given by the dynamical equation
dQ
(
t
)
=
C
+ ξ(
t
)
(3.149)
dt
in a notation consistent with the physics literature. Here the small change in net evidence
dQ given in a time increment dt is the constant Cdt
which is the deterministic increase
in evidence of the correct decision per unit time due to a constant drift. As Bogacz
et al . explain, we have C
,
>
0 if hypothesis H + is correct for the realization in question
and C
0 if hypothesis H is correct and the random-force term is usually chosen to
be white noise as it is for physical processes. We emphasize here that the choice of
statistical distribution for the random fluctuations is made for convenience, with little
or no justification in terms of experimental evidence. In fact the evidence we discuss in
subsequent chapters suggests that statistical distributions other than those usually used
in the literature for the random force might be more appropriate.
The formal solution to the dynamical equation, when Q (0)
<
=
0, is given by
t
0 ξ(
t )
dt .
Q
(
t
) =
Ct
+
(3.150)
This solution can be used to construct the characteristic function for the decision-
making process. The characteristic function is given by the Fourier transform of the
probability density p
(
q
,
t
)
, namely
e ikQ ( t )
e ikq p
φ(
k
,
t
) = FT [
p
(
q
,
t
) ;
k
]≡
(
q
,
t
)
dq
=
ξ .
(3.151)
−∞
Here it should be clear that we are evaluating the average value of the exponential
function and for this we can use the solution to the Langevin equation and thereby
obtain
e ikCt e ik 0 ξ (
dt
t
)
φ(
,
) =
ξ .
k
t
(3.152)
The average in ( 3.152 ) can be evaluated by expanding the exponential and using the
assumed statistical properties of the random force to evaluate the individual moments
in the series expansion. In particular, on treating the moments of a Gaussian random
variable factor as
x 2 n
x 2 n
=
and using the delta-correlated character of the random force, the series can be resummed
to yield for the characteristic function
e ikCt 2 Dtk 2
φ(
k
,
t
) =
.
(3.153)
 
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