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and the vector
f
0
D
f
.
v
D
0
/
and the matrix coefficients of the expansion,
M
i
,are
to be determined below.
The unusual vector notation
v
2
represents a vector of length
p
2
such that,
v
2
D
v
˝
v
D
v
1
v
T
v
2
v
T
v
p
v
T
T
;
(7.6)
where the symbol “˝” refers to the Kronecker product and
v
i
is the ith element of
the innovation vector. A similar representation applies to the
p
3
-vector
v
3
and so on.
The entire polynomial expansion in (
7.5
) may formally be represented as
x
x
f
D
f
0
C
G v
(7.7)
where
G
D
Œ
M
1
M
2
:::
M
1
;
(7.8)
v
D
v
T
v
2T
v
1
T
T
:
(7.9)
Equation (
7.7
) describes how to calculate the mean of the posterior using the
“linear” update involving the gain matrix
G
operating on the predictor vector
v
,
where
v
is comprised of an infinite number of predictors formed from today's
innovation.
To determine the vector
f
0
we note that a random draw from the posterior
distribution is
x
D
x
C ©
is a random variable with zero mean. Therefore,
an equation for the “error” in estimating the true state as the posterior mean may be
obtained by subtracting
x
from both sides of (
7.7
):
,where
©
© D ©
f
.
f
0
C
G v
/:
(7.10)
Because the expected value of
©
and
©
f
must vanish this implies that
f
0
D
G
h
v
i
;
(7.11)
where
h
i
T
h
v
i
T
˝
v
2
˛
T
h
v
i D
;
(7.12)
and the notation hi represents the expected value of a random variable. Note that if
we assume that h
v
iD
0
this implies that we have assumed that the observation and
the mean of the prior are accurate in so far as the distribution of truth about them is
unbiased.
Given (
7.11
) we may now re-write (
7.10
)as
© D ©
f
G v
0
;
(7.13)
where
v
0
D
v
h
v
i, which now clearly has the property that the “errors” have zero
mean.
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