Let’s say everything subscripted with a 1 denotes the first column of whatever quantity it is indexing, i.e. the first column of \hat{v}_{0} I’ll call \hat{v}_{0, 1}. Your model is
\hat{v}_{0, 1} = (V - \alpha_{1}) \mathop{/} \beta_{1} \\
\alpha_{1} = \alpha_{\text{raw}, 1} * \gamma_{1} \quad
\beta_{1} = (\beta_{\text{raw}, 1} + 1 + \mu_{\beta_{\text{raw}, 1}}) * \sigma_{\beta, 1} - \sigma_{\beta, 1} + 1 \\
\alpha_{\text{raw}, 1} \sim \text{N}(0, 1^2) \quad
\beta_{\text{raw}, 1} \sim \text{N}(1, 1^2) \quad
\sigma_{\beta, 1} \sim \text{Cauchy}_{+}(0, 0.5) \quad
\gamma_{1} \sim \text{Cauchy}(0, B)
for some fixed B.
Initial Questions:
- I don’t understand the transformation you use to define \beta_{1}. Can you elaborate on why you are doing this? It almost looks like some kind of non-centred parameterisation; subtracting \sigma_{\beta, 1} in particular is strange.
- Those Cauchy priors are heavy, \gamma_{1} is going to require a large amount to data to identify, and will be heavily influenced by the prior. Set up a prior predictive check pipeline to investigate the effect of those Cauchy priors in particular
Your specific question:
- You have priors for \alpha_{\text{raw}, 1}, \beta_{\text{raw}, 1}, \sigma_{\beta, 1} and \gamma_{1}, which induce a prior on \hat{v}_{0, 1}. Specifying an additional prior on \hat{v}_{0, 1}, as you do in line 70 of you Stan code, is somewhat incoherent. It could theoretically be made coherent via pooling, which @maxbiostat has talked about on these forums before here and here, but as written you are effectively “specifying two priors” for \hat{v}_{0, 1}.
- If you were to coherently specify a “joint prior” for \text{p}(\alpha_{\text{raw}, 1}, \beta_{\text{raw}, 1}, \sigma_{\beta, 1}, \gamma_{1}, \hat{v}_{0, 1}) by specifying priors for the marginals of the aforementioned prior that admit a normalised probability density for the joint prior, then I think a Jacobian correction would be necessary, as the relationship between \hat{v}_{0, 1} and \alpha_{\text{raw}, 1}, \beta_{\text{raw}, 1}, \sigma_{\beta, 1} and \gamma_{1} is non-linear, but I’m still waiting for the caffeine from my coffee to kick in, so I could be wrong there.