Modelling joint likelihood of two outcomes & parameter correlation

Hello! I’m planning to model the joint likelihood of two hierarchical outcomes — one using regression, the other using logistic regression.

y1 \sim N(k_{c[i]} + k_{u[i]} * U, \sigma_{[i]})
V = k_{b[i]} + k_{p[i]}*P
y2 \sim BernoulliLogit(C, V)

(1) Assuming there are theoretical reasons why individual-level values of k_u and k_p should be relatively strongly correlated, what would be the most sensible way of imposing prior expectations on this correlation?

And (2), as model fit should depend on the fit of the model to both y1 and y2 (which are not independent, due to the correlating parameters), is it correct to calculate log-likelihood by summing across both predictions as below?

log_lik[i] = normal_lpdf(Bid[i] | bid_mu, sigma[S[i]]) + bernoulli_logit_lpmf(Choice[i] | V[i]);

Thank you in advance for your help and sorry if these are naive questions – I’ve done my best to inform myself on this issue but couldn’t find specific answers to these questions.

Sorry this didn’t get answered earlier, @mjiwa. I’m just going through cleaning up unanswered non-bras questions.

I don’t understand your k notation. Is that one variable k and with subscript c_i and u_i? Or is it two different variables k_c and k_u?

But I guess that doesn’t matter, because we cover (1) in the User’s Guide:

(2) Yes, that’s the log likelihood for observation of the pair (Bid[i], Choice[i]).