Strictly decreasing conditional variance on
0<=q_v<=1 → weak decrease on that interval and strict
decrease only for 0<q_v<1 under a strict
prior-strength increase (Theorem 3.1 and Appendix A). Here
q_v is true quality. At q_v=0 or
1, the Bernoulli variance is zero for every prior strength,
so strict decrease fails.
Appendix D, Equation
(20): Beta prior shape
Beta(C,1) → Beta(C,1-C). With prior
strength m>0 and 0<C<1, the scaled
law is Beta(m C,m(1-C)), yielding the displayed posterior
mean (k+m C)/(n+m) after k positive reviews
among n observations.
Appendix E,
Equation (21): Dirichlet posterior average
The count-only average →
sum_j (alphaHat_j+N_j) r_j / sum_j (alphaHat_j+N_j), where
alphaHat_j are prior pseudo-counts, N_j
observed counts, and r_j rating scores. The Dirichlet
update requires the prior terms. Zero prior weight recovers the
count-only formula algebraically when the observed total is positive,
but all-zero shapes do not define a proper Dirichlet prior.