Hierarchical ordered multinomial regression

Modeling
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To answer a followup question from @CerulloE one can also model heterogeneity in the cut points with the induced Dirichlet prior.

Recall that a hierarchical model takes the form

\pi( \mathbf{c}_{n} | \phi) \cdot \pi(\phi),

where the population model \pi( \mathbf{c}_{n} | \phi) is some self-consistent model for the parameters \mathbf{c}.

Above I suggested reparameterizing the cut points, \mathbf{c}_{n} \mapsto \boldsymbol{\delta}_{n} to remove the constraint, in which case the the individual unconstrained parameters can be modeled with independent normal population models,

\pi( \boldsymbol{\delta}_{n} | \phi) = \prod_{k = 1}^{K} \pi(\delta_{n, k} | \mu_n, \tau_n).

But one can also just used the induced Dirichlet prior for the population model directly,

\pi( \mathbf{c}_{n} | \phi) = \text{induced-Dirichlet}( \mathbf{c}_{n} | \boldsymbol{\alpha}).

In other words instead of fitting a model with

\text{induced-Dirichlet}( \mathbf{c}_{n} | \boldsymbol{\alpha})

where the \boldsymbol{\alpha} are constants one can fit

\text{induced-Dirichlet}( \mathbf{c}_{n} | \boldsymbol{\alpha}) \cdot \pi( \boldsymbol{\alpha})

where the \boldsymbol{\alpha} are parameters.

For the population prior one might try something like \pi( \boldsymbol{\alpha}) = \prod_{n} \pi(\alpha_{n}) or even modeling the \log \alpha_{n} with a multivariate normal.

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