Thanks! So I can write the posterior of F for the given graphical model in the paper as
P(F_{uc}|F_{-uc},X,A)\propto p(F_{uc}|F_{-uc})\int_W P(X|F,W)P(W)dW\int_{F'} P(A|F,F')P(F')dF
In the first integral we can marginalize over W and in the second one over all F' but each of these two integral corresponds to a likelihood and the computed values of them have at least one oder of magnitude difference to the other. How can I define a scale parameter in this model and if the likelihoods and priors over W are normal distribution, would it be a problem to have a beta distribution as the prior over \alpha parameter in order to keep the conjugacy in this model?