# Model comparisons and point hypotheses

I have a quick general question about point hypotheses in models.

Say I have some predictor data vector x which predict multiple observed data y,z with measurement errors \epsilon,\delta which have some known distribution. The model has some shared parameters \theta between y and z and some parameters a,b,c,d which are not shared.

y_i=f(x_i,a,b,\theta)+\epsilon_i
z_i=g(x_i,c,d,\theta)+\delta_i

I want to know the value of P(a=c \cap b=d), i.e. the probability that the a and b parameters used to fit y are equal to the b and d parameters used to fit z. This is kind of like a point hypothesis in lower dimensional settings.

Is the only way of computing such a probability to use model comparison (as a model in which the restriction a=b \cap c=d applies is more parsimonious)? If so, what type of model comparison would be most recommended for a high dimensional problem.

Short answer: point hypothesis are hard to consolidate with the Bayesian paradigm, there are multiple ways to achieve something similar depending on your overarching goals.