Hybrid Dynamic Bayesian Network

In a dynamic bayesian network, if a discrete child has both discrete and continuous parents, how could one do the inference specially the variational inference? For instance, in the below graphical modelenter image description here
where discrete variables are circles and continuous one is a square. In the Murphy thesis (Appendix A), it was mentioned in this case one can use the conditional softmax but he did not go deeper into the topic and I did not find any paper with this type of hybrid model.

Could anybody explain or suggest a way to do the inference in this type of model? How should this conditional softmax function look like in my model?

From a quick look of the link, to get the conditional softmax you create a softmax function for each discrete state of the discrete parent(s). And then I guess you can infer the parameter in the softmax function and the probability of the discrete state jointly.