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7.4.2
Mean and Variance
Given the predictive density, point estimates and information about the predic-
tion confidence are given by its mean and variance, respectively. As the mixture
of Student's t distributions might be multi-modal, there exists no clear definition
for the 95% confidence intervals, but a mixture density-related study that deals
with this problem was performed by Hyndman [117]. Here, the variance is taken
as a sucient indicator of the prediction's confidence.
Let us first consider the mean and variance for arbitrary mixture densities,
and subsequently apply it to (7.108). Let
{
X
k
}
be a set of random variables
to give
X
=
k
g
k
X
k
.Asshown
by Waterhouse [227], the mean and variance of
X
are given by
that are mixed with mixing coecients
{
g
k
}
(
X
)=
k
var(
X
)=
k
(
X
k
)
2
)
(
X
)
2
.
(7.109)
E
g
k
E
(
X
k
)
,
g
k
(var(
X
k
)+
E
−
E
The Student's t distributions in (7.108) have mean
w
kj
T
x
and variance (1 +
x
T
Λ
k
−
1
x
)2
b
τ
k
/
(
a
τ
k
−
1). Therefore, the mean vector of the predictive density
is
x
,
X
,
Y
)=
k
x
,
(
y
|
g
k
(
x
)
|
v
k
=
v
k
W
k
E
(7.110)
and each element
y
j
of
y
has variance
var(
y
j
|
x
,
X
,
Y
)
(7.111)
2
1
(1 +
x
T
Λ
k
−
1
x
)+(
w
kj
T
x
)
2
=
k
b
τ
k
a
τ
k
−
g
k
(
x
)
|
v
k
=
v
k
(
y
|
x
,
X
,
Y
)
j
,
−
E
x
,
X
,
Y
).
These expressions are used in the following chapter to plot the mean predic-
tions of the LCS model, and to derive confidence intervals on these predictions.
(
y
|
x
,
X
,
Y
)
j
denotes the
j
th element of
(
y
|
where
E
E
7.5
Model Modifications to Perform Classification
In order to adjust the Bayesian LCS model to perform classification rather than
regression, the input space will, as before, be assumed to be given by
D
X
.
X
=
R
D
Y
,where
D
Y
is the number
of classes of the problem. For any observation (
x
,
y
), the output vector
y
defines
the class
j
associated with input
x
by
y
j
= 1 and all other elements being 0.
The task of the
The output space, on the other hand, is
Y
=
{
0
,
1
}
LCS
model for a fixed model structure
M
is to model the
probability
p
(
y
) of any class being associated with a given input. A good
model structure is one that assigns high probabilities to a single class, dependent
on the input, without modelling the noise.
|
x
,
M
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