We note that the set of parameters $\quad \theta = (q_{k},\Sigma^{(k)} )_{k \in [K]} $ where $\Sigma^{(k)}$ is a SDP Matrix (symmetric definite matrix)
We have also $\quad \mathbb{P}[z_{u}=k]=q_{k}\quad$ with $\quad q_{k}\geq0 \quad$ and $\sum_{k \in [K]} q_{k}=1 \quad and \quad z_{u}$ is random variable (type of $u\in [P]$) wich has values in [K]
as $u \in [P],\quad q_{k}*P \quad$is the number of u such that $z_{u}=k \quad$and with $\quad \mathbb{P}_{\theta} (rep. \mathbb{E}_{\theta})\quad $the conditional probability for the parameters
given these information below :
- $(N_{uj}|(z_{u}=k,(\alpha_{kj})_{kj}) \quad \sim \quad Poisson(exp(\alpha_{kj}))$
- $(N_{uj}|(z_{u}=k,(\alpha_{kj})_{kj}))_{j\in [J]} \quad $ are independant and K vectors (in dimension J) and $\quad (\alpha_{kj}:j\in [J])_k\in[K]\quad$ are independant and for all $k \in[K] \quad (\alpha_{kj})_{j \in [J]}\quad \sim \quad\mathcal{N}(0,\Sigma^{(k)})$
- $\mathbb{P}[z_{u}=k]=q_{k}\quad$
- $N_{uj} > 0 \quad \forall u \in [P] \quad and \quad \forall j \in [J]$
we are willing to find the following function
$R_{\theta}((c_{j})_{j=1}^{J})=\frac{1}{P} \sum_{u_{0} \in[P]} \mathbb{P}_{\theta}(\sum_{j \in [J]} N_{u_{0}j}>0 | \sum_{u \in [P]}N_{uj}=c_{j}\quad \forall j \in [J])$
I tried to calculate it but the result does not depend on $\theta$, can somoene tell me how can I calculate this with given informations