Consider a decision maker (DM) who as to choose an action from the finite set $\mathcal{Y}$ with cardinality $L. $
The payoff the he gets depends on the action chosen and the state of the world. The state of the world is denoted by $z\equiv (z_1,...,z_L)\in \mathcal{Z}^L\subseteq \mathbb{R}^L$. The payoff when the DM chooses action $y\in \mathcal{Y}$ and the state of the world is $z\in \mathcal{Z}^L$ is denoted by $u(y,z_y)$.
Before choosing the DM does not observe the state of the world. His prior on $z$ is denoted by $F_Z\in \Delta(\mathcal{Z}^L)$ and has marginals that are equivalent across $y$.
Before choosing the DM can discover $z$ by searching sequentially and paying cost $c_y$ when uncovering $z_y$ for each $y\in \mathcal{Y}$.
When search stops, the DM chooses action $y$ maximising his payoff among the searched actions.
Question
Can we find a conditional signal density $P_{T}(\cdot|z)\in \Delta(\mathcal{T})$ such that the problem above can be rewritten as $$ \max_{y\in \mathcal{Y}}\sum_{z\in \mathcal{Z}} u(y,z_y) \underbrace{\frac{P_T(t|z) F_Z(z)}{\sum_{z\in \mathcal{Z}}P_T(t|z) F_Z(z)}}_{\text{Posterior belief by Bayes rule}} $$ for each $t\in \mathcal{T}$?