16.322 Stochastic Estimation and Control, Fall 2004
Prof. Vander Velde
Page 1 of 8
Lecture 22
Last time:
1
1
1
22
1
ˆ
1
ˆ
ˆ
ˆ
11
N
k
kN
xk
N
x
z
x
σ
=+
+
=
+
∑
∑
The estimate of
x
based on
N
data points can then be made without
reprocessing the first
1
N
points.
Their effect can be included simply by starting
with a pseudo observation
which is equal to the estimate based on the first
1
N
points having a variance equal to the variance of the estimate based on
1
N
.
The same is true of the variance of the estimate based on
N
.
1
1
2
1
ˆˆ
1
N
xx
k
σσ
∑
A priori information about
x
can be included in exactly this way whether or not
it was derived from previous measurements.
Whatever the source
of the prior
information, it can be expressed as an a priori distribution
()
f x
, or at least as an
expected value and a variance.
Take the expected value as a pseudo observation,
2
0
, and accumulate this data with the actual data using the standard formulae.
With the prior information included as a pseudo observation, the least squares
estimate is formed just as if there were no prior information.
The result, for
normal variables at least, is identical to the estimators based on the conditional
distribution of
x
.
Bayes’ rule can be used to form the distribution
( )
12

,
,...,
N
fxzz
z
starting from
the original a priori distribution
f x
1
1
1

( )


( , )
etc.
fx
fxfz x
fxz
fuf z ud
u
f xz f z x
fuz f z ud
u
=
=
∫
∫
if the measurements are conditionally independent.
Two disadvantages relative
to the previous method:
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Prof. Vander Velde
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•
More computation unless you know each conditional density is going to be
normal
•
Must provide
()
f x
 the a priori distribution.
This is both the advantage and
disadvantage of this method.
Other estimators include the effect of a priori information directly.
Several
estimators are based on the conditional probability distribution
of
x
given the
values of the observations
.
In this approach, we think of
x
as a random variable
having some distribution.
This troubles some people since we know
x
is in fact
fixed
at some value throughout all the experiment.
However, the fact that we do
not know what the value is is expressed in terms of a distribution of possible
values for
x
.
The extent of our a priori knowledge is reflected in the variance of
the a priori distribution we assign.
Having an a priori distribution for
x
, and the values of the observations, we can
in principle – and often in fact – calculate the conditional distribution of
x
given
the observations.
This is in fact the a posteriori distribution,
( )
1

,...,
N
f xz
z
.
This
distribution expresses the probability density for various values of
x
given the
values of the observations and the a priori distribution.
Having this distribution,
one can define a number of reasonable estimates
. One is the minimum variance
estimate
– that value
ˆ
x
which minimizes the error variance.
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 Fall '04
 WallaceVanderVelde
 Normal Distribution, Variance, Probability theory, Estimation theory, Prof. Vander Velde

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