quantum mechanics - What is the physical meaning of weak expectation values?

In the two-state formalism of Yakir Aharonov, the weak expectation value of an operator $A$ is $\frac{\langle \chi | A | \psi \rangle}{\langle \chi | \psi \rangle}$. This can have bizarre properties. If $A$ is Hermitian, the weak expectation value can be complex. If $A$ is a bounded operator with the absolute value of its eigenvalues all bounded by $\lambda$, the weak expectation can exceed $\lambda$.

If $A = \sum_i \lambda_i P_i$ where $\{P_i\}_i$ is a complete orthonormal set of projectors, the strong expectation value is

$$\frac{\sum_j \lambda_j |\langle \chi |P_j | \psi \rangle|^2}{\sum_i | \langle \chi | P_i | \psi \rangle |^2}$$ which is also confusing as the act of measurement affects what is being postselected for.

More specifically,

$$|\langle \chi |\psi\rangle|^2 = \sum_{i,j} \langle \psi | P_i| \chi \rangle \langle \chi | P_j | \psi \rangle \neq \sum_i |\langle \chi |P_i|\psi \rangle |^2$$ in general.