| s_i \rangle \quad , \quad  \sum_j \, \langle b_j | s_i \rangle = 1.$$ $$r^{(t)}= \frac{1}{N} \sum_{i=1}^N \sum_{j=1}^{N'} \, \langle a_{s_i}^{*} | \, B_j \, | s_i \rangle = \frac{1}{N} \sum_{i=1}^N \sum_{j=1}^{N'} \, \langle a_{s_i}^{*}  | b_j \rangle \, \langle b_j | s_i \rangle
\label{blockeff}$$ where $ \langle \alpha | \beta  \rangle = P^{t} ( \alpha | \beta) $ and $ {N'} $ stands for the number of possible belief states which in immediate examples equal to the number of world states $ N $. One can also write the belief states’ vectors with respect to the world states using an inverse matrix for probabilities $\langle b_i | s_i \rangle $. Notice that in the probability matrix of $\langle b_k | s_k  \rangle $, the summation of elements on a row is equal to one $ \sum_j \, \langle b_j | s_i \rangle = 1 $ but it is not the case for the summation on a column $ \sum_j \, \langle b_i | s_j \rangle $. We can also, assume that $${\delta}_{ij} = \langle s_i | s_j \rangle \, = \langle a_i | a_j \rangle \ = \langle b_i | b_j \rangle$$ where needed.

As a more general 