We have a probabilistic model with two parameters, $\theta$ and $\eta$, both of which are uniformly distributed between 0 and 1. The model has five possible outcomes, and the probability of each outcome is given by:

$$P(y=i|\theta,\eta) = \begin{cases} \theta/4 + 1/8, & i=1 \\ \theta/4, & i=2 \\ \eta/4, & i=3 \\ \eta/4 + 3/8, & i=4 \\ 1/2*(1-\eta-\theta), & i=5 \end{cases}$$

We have observed 22 trials of the experiment, and the number of times each outcome occurred is given by the vector $\mathbf{y} = (14, 1, 1, 1, 5)$. Our goal is to use Gibbs sampling to estimate the values of $\theta$ and $\eta$ that are most likely to have generated the observed data.

The original idea is to assume that the parameters fllow the Dirichlet distribution, so the posterior distribution also has a Dirichlet distribution, but when calculating the conditional distribution of parameter $\theta$ with respect to parameter $\eta$ and $y$, $f(\theta | \eta,\mathbf{y})$,it encounters difficulties, and naturally it is impossible to perform sampling updates.Any help from you is greatly appreciated.

  • $\begingroup$ Since$$1-\eta-\theta\ge0$$this means $\theta$ and $\eta$ cannot be uniformly generated. $\endgroup$
    – Xi'an
    Aug 29 at 20:28


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