Ah, I wasn’t sure which details would be useful… In essence, each of the 1024 possible values of the categorical variable l corresponds to a set of ordered pairs:
l = \{(m, u)\, \in \mathbb{N} \times \mathbb{N}\}
And to a set of sets in the case of t:
t = \{c_i \, | ,\ c_i = \{m \in \mathbb{N} \}\}
So the likelihood depends on what these sets look like:
P((m,u)=D_{ki}|l,t)\propto \sum_{c_i\in t}{\frac{L(m|u,c_i,l)}{|t|}}
L(m \,|\, u,c_i,l)= \left\{
\begin{array}{@{}rl@{}}
& g(u,l,c_i), \quad (m, u) \in l \quad and \quad m \in c_i; \\
& h(u,l,c_i), \, \, \, \, \, \, (m, u) \notin l \quad and \quad m \in c_i;\\
& 0, \quad \, \, \, \, \, \, \, \, \, \, \, \quad \quad m \notin c_i\\
\end{array}
\right .
Where g is based on counting the number of pairs in l of the form (n, u), n \in c_i; h is chosen so everything can add up to 1.
But in a more general case, I am quite confused about what steps I would need to follow to get to (or estimate) the marginals Q^k(l|D_1,...,D_k) and Q^k(t_k|D_1,...,D_k) from my joint posterior Q^k(l,t_1,...,t_k,\theta|D_1,...,D_k).
Sorry if this thread is getting too long and I really appreciate the help so far!