Probabilistic Forms

Likelihood is:

$$p(Y\vert T,S,\theta) = p\left( \epsilon = Y - G(T(S)) \right)$$

$$= \left( \frac{\beta}{2 \pi} \right)^{N/2} \exp\left( \frac{-\beta \Vert Y- G(T(S))\Vert^2}{2} \right)$$

Priors:

$$p(\beta) = \frac{1}{\beta}$$

For $\alpha$ there are two alternative priors that are useful.

1. Flat prior:
   

   $$p(\alpha_j) = \left\{ \begin{array}{ccl} C_1 & \qquad & \mathrm{for~}0 \le ... ...j \le C_1^{-1} \\ 0 & & \mathrm{otherwise} \end{array} \right.$$

   
   
   where the range of $\alpha_j$ is restricted to $[0,C_1^{-1}]$.
2. Or a prior encoding prior knowledge:
   

   $$p(\alpha \vert \lambda, Q ) = \vert\det(Q)\vert^{1/2} \left(\frac{\lambda}{\pi}\right)^{D/2} \exp\left( - \lambda \alpha^{\mathrm{\textsf{T}}}Q \alpha \right)$$

   $$p(\lambda) = C_0$$

   
   
   where $Q$ represents the prior knowledge about the expected intensities in the image formation; $\lambda$ is a parameter expressing the unknown scaling between the learnt prior distribution of $\alpha$ parameters and the intensities in a new image; and $C_0$ is a constant, representing the (improper) flat prior on $\lambda$.

Parameters:

$$\theta = \{ \beta, \alpha , \lambda \}$$

where $\beta = \frac{1}{\sigma^2}$ is the precision parameter.