Worked connection for MMP-INTERVENTION-AND-INFORMATION

1. Establish what the actions would do. A service must choose response A or B. Its current circumstance is H0 or H1, each with supplied probability 0.5. Earlier randomized work recorded the circumstance and supplies the following population mean losses:

MMP.15 makes the use of these records explicit. The case assumes consistent response versions, no interference, positive assignment probability for each response in each recorded circumstance, and transfer of the conditional intervention means to the current service. Under those assumptions, randomization identifies the means in the table. They are stipulated population quantities here; estimates from finite records would also need MMP.13’s uncertainty.

Without another indication, A has expected loss 5 and B has expected loss 2. This supplies the comparison that any proposed observation must improve.

2. Construct the record the proposed observation would supply. A diagnostic report is positive with probability 0.8 in H1 and 0.2 in H0. It does not change the circumstance or the subsequent action losses. Within each circumstance, the report supplies no further information about response loss. Its cost is 0.2 in the same loss units, and the report arrives before the response is chosen. MMP.7 retains this reporting law, rather than treating a positive report as certain knowledge of H1.

3. Compare information through its consequence. MMP.16 obtains a posterior probability of H1 equal to 0.8 after a positive report and 0.2 after a negative one. Choose B after positive: its conditional expected loss is 0.8. Choose A after negative: its conditional expected loss is 2. Each report has probability 0.5, so expected action loss is 1.4. The information reduces that loss by 0.6; including its 0.2 cost gives 1.6 rather than 2.

4. Construct the continuing instruction. MMP.8.SD represents the first choice, whether to obtain the report, and the later response choice. For the response choice after the report, the posterior belief is sufficient: the conditional mean losses depend only on H and there are no later choices. The resulting instruction is to obtain the report, then use the branch above. Subject and Method Engineering work supply the actual observing and acting capabilities. If the report cannot arrive in time, formulate the response choice from the information that will actually be available.

5. Revise only what changed. Suppose the available diagnostic now has the same positive probability, 0.5, in both circumstances. MMP.7’s new law leaves the belief at 0.5 after either report. MMP.16 returns zero information benefit for the response decision; MMP.8.SD chooses B without the diagnostic and avoids its cost. If observing instead changes the circumstance, include that transition and the changed intervention consequences before reusing the former calculation.

6. Recover the comparison if the question changes. A later inquiry may ask which cases would have benefited from the other response, rather than which response minimizes future expected loss. MMP.19 constructs the relation between a case’s alternative responses. For an observed case, MMP.7/.13 first recover what its factual record implies about the underlying conditions; C.28.MR then changes the mechanism while retaining that case information.

Check what the available laws actually determine. In MMP.19:5.2, two models have identical action-specific success probabilities and identical randomized records, but give different answers about the same observed case under the other action. MMP.19 returns that ambiguity; a feasible observation is considered through MMP.16 only if it could resolve a useful distinction. Returning to the future expected-outcome criterion can make the unresolved same-case relation irrelevant. The earlier choice is then usable without that extra inquiry.

The connection can stop at a sufficient existing choice. The needed comparison determines whether to construct a report, a continuing instruction or a same-case counterfactual.