Let $X$ be the child variable of the distribution, and assume that
it can take on $M$ possible states labeled $x_1$ through
$x_M$ in increasing order. The generalized partial
credit model defines a set of functions
$Z_m(\theta_1,\ldots,\theta_K)$ for $m=2,\ldots,M$, where
$$Pr(X >= x_m | X >=x_{m-1}, \theta_1,\ldots,\theta_K) = logit^{-1}
-D*Z_m(\theta_1,\ldots,\theta_K)$$
The conditional probabilities for each child state given the effective
thetas for the parent variables is then given by
$$Pr(X == x_m |\theta_1,\ldots,\theta_K) \frac{\sum_{r=1}^m
Z_r(\theta_1,\ldots,\theta_K)}{\sum_{r=1}^M
Z_r(\theta_1,\ldots,\theta_K)}$$