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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?

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  • $\begingroup$ Are the outcomes exclusive, dependent, independent? $\endgroup$ – Tim Nov 30 '15 at 14:37
  • $\begingroup$ Exclusive (meaning only one can occur?) and independent $\endgroup$ – sammosummo Nov 30 '15 at 15:03
  • $\begingroup$ Yes. Could you edit so the description of your data is more clear? $\endgroup$ – Tim Nov 30 '15 at 15:05
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You may be interested in some variation on the basic Dirichlet/categorical model, described as:

\begin{align*} p & \sim \text{dirichlet}(\alpha) \\ x \, | \, p & \sim \text{categorical}(p) \end{align*}

This is a generalization of a standard conjugate Bayesian model for binary outcomes.

As an example, consider a symmetric Dirichlet prior with concentration parameter $\alpha_0 = (1, 1, 1)$ and observe a sample of observations $1, 1, 1, 2, 1, 2, 2, 0, 2, 2$. Due to conjugacy, the posterior distribution is also a Dirichlet distribution with concentration parameter $\alpha = (2, 5, 6)$, and so you get that the posterior mean is $\left(\frac{2}{13}, \frac{5}{13}, \frac{6}{13}\right)$.

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