# Transformation of cutpoints in Ordinal Probit/Logit regression

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 -

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