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In this work we use a Bayesian framework. Equations 1
and 2 form our likelihood. The distribution we are
interested in is the full posterior distribution over the model parameters,
and depends upon this
likelihood and the priors over the unknown parameters in our model
via Bayes rule:
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(3) |
We now need to consider the specification of priors over the
parameters in our model. A priori we assume independence between
the parameters:
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(4) |
As we shall see, using independent priors allows us to use
conjugate priors, which in turns makes the model tractable when
using Variational Bayes. However, assuming that we have
independence between priors for different parameters does not mean
that the parameters will be independent in the posterior. Any
dependence between parameters inferred from the data and the
likelihood will still be reflected in the joint posterior. For the
precision we assume a voxelwise noninformative Gamma prior:
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