| dbp:proof
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- Plugging back, and simplifying, we have (en)
- We know is a gaussian, and is another gaussian. We also know that these are independent. Thus we can perform a reparameterization: where are IID gaussians.
There are 5 variables and two linear equations. The two sources of randomness are , which can be reparameterized by rotation, since the IID gaussian distribution is rotationally symmetric.
By plugging in the equations, we can solve for the first reparameterization: where is a gaussian with mean zero and variance one.
To find the second one, we complete the rotational matrix:
Since rotational matrices are all of the form , we know the matrix must be and since the inverse of rotational matrix is its transpose, (en)
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