This is a very general question concerning Bayesian model design. I would like to construct a model that predicts a subject's performance in an experiment. The model will have a number of free parameters that influence the subject's decision on each trial, and I want to estimate those parameter values via Bayesian inference.
If the experiment has a binary outcome on each trial (eg was the trial correct or incorrect), I would code these as a vector of Bernoulli random variables, with input p being determined by the free parameters in some way. However, what if I have four independent possible response outcomes per trial that at not ordinal? What probability distribution should be assigned to the observations in this circumstance?