Apologies in advance if this is the wrong place to be posting this! I’m trying to run some prior predictive checks for the linear predictor portion of a basic multiple regression. With simple regression (one predictor) of the following form:

I understand how to sample values of alpha and beta from the priors to fit various values of mu as a function of a sensible range of X values.

I’m unsure of how this would work with a multiple regression of the form:

Would it make sense to chunk out values of (alpha and beta_1) and (alpha and beta_2) to create two plots of possible lines against possible X values, one for each predictor? Or would this no longer make sense given the implied interdependencies between all three coefficients, in which case I should be examining the distribution of computed mu values or some other quantity?

Thank you!