# Priors for compositions

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#### 2024-01-24

Let’s load the necessary packages:

    library(zoid)

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()`.

    fit <- fit_zoid(data, prior_sd = 2)

## 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)