This formulation allows convenient transformations of distributions
through other group elements using the adjoint. Consider ,
and the distribution
with mean
. Consider
where
is a sample
drawn from
:
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(217) |
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(218) |
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(219) |
By the definition of covariance,
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(220) |
Given a linear transformation , by linearity of expectation we
have:
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(221) |
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(222) | |
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(223) |
So we can express the parameters of the transformed distribution:
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(224) |
Thus the mean of the transformed distribution is the transformed mean, and the covariance is mapped linearly by the adjoint.
Ethan Eade 2012-02-16