Let’s load the necessary packages:
This vignette is designed to illustrate how to build priors for the
estimated compositions, \(\textbf{p}\).
When covariates are included in the design matrix, all elements of \(\textbf{p}\) in the Dirichlet regression
(both intercepts and slopes or offsets) are assigned \(\sim N(0,1)\) priors. This approach is
similar to the improper priors used in brms::brm(). These
priors may be weakly informative, and the user may wish to change the
standard deviation – which can be done with the prior_sd
argument in fit_zoid().
A more familiar approach may be to work with Dirichlet priors. We can
adjust the standard deviation in our Normal priors to match the
Dirichlet. The helper function for this uses optim to
minimize the RMSPE between the observed and target values. For example,
if we had 8 bins and wanted to find the Dirichlet prior that would
correspond to hyperparamters \((\alpha)=1\), we could call the
fit_prior function.
The sd object is a list that contains (1) the estimated
standard deviation, (2) the value of the objective function at
convergence, and (3) whether or not convergence occurred (anything other
than 0 is problematic). The value of the standard deviation here in
sd$sd is 1.200453.
So in this case, a standard deviation of ~ 1.20 yields a prior
equivalent to a \(\sim Dirichlet(1)\)
prior. This new value can then be entered into our model with the
prior_sd argument,