Undisclosed qubit measurement outcome with an unspecified preparation prior ($\rho =\mathrm{diag}(q,1-q)$)

After a computational-basis projective measurement, an observer who lacks the result assigns a diagonal density matrix. If the observer's preparation prior has mean $q=\mathbb{E}|\alpha {|}^2$, the assignment is the displayed matrix. No outcome record and no stated prior do not logically force $q=1/2$; that value follows from a symmetric or maximum-entropy ignorance convention. An observer who knows the preparation instead uses $q=|\alpha {|}^2$.