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()`

.

## Dirichlet priors

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.

```
set.seed(123)
sd = fit_prior(n_bins = 8, n_draws = draws, target = 1, iterations=iter)
```

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,

`fit <- fit_zoid(data, prior_sd = 1.2)`