In 't Hooft Cellular Automation Interpretation it is declared, that for system described by some template state $\lvert \psi \rangle$ and for any ontological measure outcome $\langle a \rvert$ the absolute square of inner product $\langle a | \psi \rangle$ is a probability to find system in $\langle a \rvert$ state.

But if we remember, that classical states are not just ontological states, but sets of ontological states — because classical, macroscopic states do not describe state precisely, then we can face the following difficulty.

Consider macrostate $X$, which can be realized in one of many ($N$) sub-microscopic ontological states $\langle a_i \rvert$, $i = 1 \ldots N$. Suppose we know that current system state can be described using $\lvert \psi \rangle$ template state. Then if we want to calculate probablity to find the system in $X$ state, we should compute $\sum_{i=1}^N \lvert \langle a_i | \psi \rangle \rvert^2$. But this is not how we do real computations in quantum mechanics. In quantum mechanics we describe classical outcome with some pure state $\langle X \rvert$ and just compute one projection $\langle X | \psi \rangle$.

So my question is: how $\sum_{i=1}^N \lvert \langle a_i | \psi \rangle \rvert^2$ becomes $\lvert \langle X | \psi \rangle \rvert^2$ and what is $\langle X \rvert$ in 't Hooft interpretation?