Across 40 sector desks the true annual alphas are distributed N(0, 2^2) in percent. One desk reports 6.0% with a standard error of 3.0%. Give the posterior mean for that desk's true alpha, the weight placed on its own number, and the posterior standard deviation.

Across 40 sector desks the true annual alphas are distributed N(0, 2^2) in percent. One desk reports 6.0% with a standard error of 3.0%. Give the posterior mean for that desk's true alpha, the weight placed on its own number, and the posterior standard deviation.

Approach: Use the normal-normal conjugate update, which averages prior and observation with weights equal to their precisions.

24/13. With prior alpha ~ N(0, tau^2) at tau^2 = 4 and likelihood y | alpha ~ N(alpha, s^2) at s^2 = 9, the normal-normal conjugate update gives a posterior mean that is the precision-weighted average (0/tau^2 + 6/s^2)/(1/tau^2 + 1/s^2) = 6*tau^2/(tau^2 + s^2) = 6*4/13 = 24/13 = 1.85%. The weight on the desk's own number is 4/13 = 0.31, so 69% of the reported 6% is shrunk away as noise. The posterior standard deviation is sqrt(1/(1/4 + 1/9)) = sqrt(36/13) = 1.66. That is tighter than either input. Shrinkage of this size is the correct response to an estimate whose standard error exceeds the cross-sectional spread of true alphas, and reporting the raw 6% would rank that desk far above peers on evidence that does not support the ordering.

Follow-up: The 40 desks have different standard errors, from 1% to 6%. What does the ranking of posterior means look like against the ranking of raw alphas?

Key concepts: normal-normal conjugacy, shrinkage, precision weighting, posterior mean.