The likelihood for an ordinal probit/logit regression model is given as -

$f(y|\beta ,\gamma ,z ) = \prod_{1}^{n} \left [ \Phi (\gamma _{j} - x_{i}'\beta ) - \Phi (\gamma _{j-1} - x_{i}'\beta ) \right ]$

While sampling in a Bayesian Setting, the problems arising due to a strict ordering of the cut points $\gamma $ can be dealt with using the transformation -

$\delta_{j} = ln \left ( \gamma_{j} - \gamma_{j-1} \right )$ for $ j = 2 ...m-1$ and $\delta_{1} = ln \gamma _{1}$ (m is the number of categories of the ordinal dependent variable).

I need to find out the log likelihood of the data conditioned on the variable $\delta$ to obtain full conditionals for Gibbs Sampling -

$ln f(y|\beta,\delta,z)$


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