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Here we give the update equation for the MRF precision parameter distribution
:
we define
as:
Even though the matrix is very sparse the
computation of the
Trace term can be very expensive to compute.
In particular, this is because
and
is a matrix whose inverse would be very
computationally expensive to compute. Therefore,
instead of computing this inverse we can compute in the linear equation:
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(46) |
Since is a positive symmetric definite matrix we can take advantage of
the conjugate gradient techniques described in Golub and Van Loan (1996). At each iteration of the conjugate gradient search for
we only need to perform one
matrix multiplication of . The conjugate gradient approach
is far
quicker than solving for the inverse of and then
multiplying by .
The conjugate gradient technique takes in
an initial guess of . Hence, as we iterate through the Variational Bayes
updates of our approximate posterior distributions, we can store the
the value of from the previous conjugate gradient solution from
the previous update of
,
and use it as the initialisation of the conjugate gradient search for
at the next update of
. After the first Variational
Bayes iteration this makes subsequent conjugate gradient
searches very quick to converge.
Next: Noise precision updates
Up: Variational Bayes Updates
Previous: Autoregressive parameter updates