Biology Reference
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invariants are not unique. However, all maximal invariants are equiv-
alent in the sense that their sets of constancy coincide.
Let
X
be distributed according to a probability distribution
P
-
,
-
.
Denote by
gX
the random variable that takes on the value
gx
when
X
x
,
and suppose that when the distribution of
X
is
P
-
, the distribution of
gX
is
P
-'
, with
-
'
also in
. The element
-
'
associated with
-
in this manner
will be denoted by
g
-
. The transformation
g
of
onto itself, defined in
this manner, is one to one provided the distributions
P
-
corresponding to
different values of
-
are distinct. The action of the group
G
induces a
partition of the parameter space
into equivalency classes.
Invariance, by reducing the data to a maximal invariant statistic
T
,
typically also shrinks the parameter space. Let
T
(
x
) be a maximal
invariant under
G
and let the distribution of
T
(
x
) depend on
v
(
-
)
. Then
the inverse image of
v
(
-
)
partitions the original parameter space into
equivalency classes. All the points that belong to the same equivalen-
cy class map onto
v
(
-
)
and conversely, if two parameter values
-
1
and
-
2
are such that
v
(
-
1
)
v
(
-
2
)
, then
-
1
and
-
2
belong to the same equivalen-
cy class. The partition of
under the group
G
and the partition induced
by
v
(
-
)
do not necessarily match each other.
We illustrate these ideas using a simple example involving only the
translation group. Let
X
i
be a bivariate random variable and let
X
i
~N
(
R
2
and
is any 2 x 2 real, symmetric, positive definite matrix}. Let the
group action consist of translations only. That is,
gX
i
,
)
. Then the parameter space
is given by
{(
,
):
X
i
1
t
i
. It then
)
where
1
is a 2 x 1 vector of 1's,
t
i
is a real
number and
is a 2 x 2 real, symmetric, positive definite matrix. The
orbit corresponding to any given
(
follows that
gX
i
~
N
(
1
t
i
,
G
is given by
*
,
*
)
under the group
*
}
.
A maximal invariant under the translation group in the above sit-
uation is given by:
Y
i
{(
,
)
:
1
t
,
(
X
i
2
X
i
1
)
. The distribution of this maximal
invariant is
N
(
2
1
,
11
22
2
12
)
. Let us denote
2
1
and
o
/
. Then, given the values of
(
,
o
/
)
, the inverse image in
the original parameter space
is given by
{(
11
22
2
12
,
)
:
2
1
,
11
o
/
}
. This defines a partition of
which is different than the
partition defined by the group
22
2
12
G
. Notice that the partition defined by
X
i
1
)
is more coarse than the partition
defined by
G
. If invariance is used, only the partitions defined by
v
(
-
)
are identifiable. This is the cost we pay due to the presence of nuisance
parameters.
Now let us consider the landmark coordinate data problem. The land-
mark coordinate matrix is denoted by
X
. Recall that the group of
the maximal invariant
Y
i
(
X
i
2
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