# How to set priors over weights for adjusting the magnitudes of different likelihoods in a model

I have a model which has two sets of observed data, e.g. X and Y. In the model, \Theta is the set of free parameters. If one has a similar setting as this paper in section “CESNA hyperparameter settings” and wants to use the bayesian inference to compute the posterior of F (for instance the parameter given in the example), which kind of prior should put over \alpha (scaling parameter or weight of the likelihoods) in order the adjust the magnitude of two likelihoods? when the likelihoods are the normal distributions and the posterior and prior over the free parameters of the model are conjugate should the conjugacy be taken into account for the prior over \alpha? In my model, one part of observational data would have greater effect with its likelihood on the sampling than the other. I am confused whether it is the only way I can tackle the sampling problem of scaling two likelihoods? I’ll appreciate for any suggestion.