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Bayesian framework

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:
$\displaystyle p(\beta,a,\phi_{\epsilon}\vert y) \propto \prod_i
p(y_i\vert\beta_i,a_i,\phi_{\epsilon_i})p(\beta,a,\phi_{\epsilon})$     (3)

We now need to consider the specification of priors over the parameters in our model. A priori we assume independence between the parameters:
$\displaystyle p(\beta,a,\phi_{\epsilon}) = p(\beta)p(a)p(\phi_{\epsilon})$     (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:
$\displaystyle p(\phi_{\epsilon})$ $\displaystyle =$ $\displaystyle \prod_i p(\phi_{\epsilon_i})$ (5)
$\displaystyle \phi_{\epsilon_i}$ $\displaystyle \sim$ $\displaystyle Ga(b_{\epsilon_0}, c_{\epsilon_0})$ (6)


next up previous
Next: Autoregressive Parameters Spatial Prior Up: Model Previous: Model