MMP.16:5.2 - A useful signal is still too costly
A service team must choose one of two recovery procedures. It models two possible failure modes H0 and H1 with current probabilities 0.5 each. The loss is remaining recovery time, in hours:
| Procedure | H0 | H1 |
|---|---|---|
| A | 0 | 10 |
| B | 4 | 0 |
With current information, expected losses are 5 for A and 2 for B. The team chooses B.
A proposed diagnostic produces either “+” or “-”. Its supplied law is P(+ | H1)=0.8 and P(+ | H0)=0.2. Each signal has marginal probability 0.5. Bayes conditioning gives P(H1 | +)=0.8 and P(H1 | -)=0.2.
After “+”, A has expected loss 8 and B has 0.8, so the team chooses B. After “-”, A has expected loss 2 and B has 3.2, so it chooses A. Expected loss with the signal, excluding diagnostic cost, is therefore:
R(d) = 0.5*0.8 + 0.5*2 = 1.4 hours.
R0 - R(d) = 2 - 1.4 = 0.6 hours.
If obtaining, interpreting and waiting for the diagnostic adds 0.7 hours to recovery, its total expected loss is 2.1 hours; using current information is better under these conditions. At an all-inclusive cost of 0.2 hours, the diagnostic gives 1.6 hours and improves the choice. The example assumes the diagnostic does not change the failure mode or the two procedures.
A different test might reveal a device identifier perfectly. If that identifier is independent of the failure mode and losses in this model, its information gain about the identifier does not reduce this recovery loss. Selecting all available uncertainty as the target would misdirect the design.
Now change the current probability of H1 to 0.95. The “-” posterior is 0.19/(0.19+0.04), approximately 0.826, and the “+” posterior is 0.76/(0.76+0.01), approximately 0.987. B remains the better procedure after either result. The diagnostic still changes beliefs about the mode, but its sample-information value for this particular choice is zero.
A later method-development investigation could use the same signal for another target, such as learning how failure modes arise. That research value needs its own question and horizon; the zero above applies to the specified recovery choice.