MMP.13:5.3 - Infer the rate for the receiving workload

Two classes of requests have different probabilities of finishing by a deadline. For class A, seven of eight observed requests finish; for class B, one of eight finishes. Assume fixed sample counts, independent Bernoulli outcomes within each class, independent class data, and independent uniform priors for p_A and p_B.

The posterior densities are proportional to p_A^7*(1-p_A) and p_B*(1-p_B)^7: Beta(8,2) and Beta(2,8). Their means are 0.8 and 0.2, and each variance is 4/275. The posteriors are independent under the stated construction.

For a future workload selecting the two classes equally, the conditional completion probability is q_old=(p_A+p_B)/2, with posterior mean 0.5. Now change only the receiving workload: its known class proportions are one-quarter A and three-quarters B. The target becomes

q_new=p_A/4+3*p_B/4.

Its posterior mean is 0.35 and its posterior variance is

(1/4)^2*(4/275)+(3/4)^2*(4/275)=1/110.

No new fitting is needed for this changed target. Carrying forward 0.5 would answer for the old mixture. If the target were an individual future completion indicator, its posterior predictive probability would be 0.35 and its variance 0.35*0.65, rather than 1/110.

The transfer assumes that the within-class probabilities remain applicable. Changed operating conditions require their own relation. Uncertain class proportions or a shared influence on the two probabilities require joint uncertainty, rather than the independent weighted-variance calculation above.