MMP.8.SD:5.2 - Pay for information only when a later choice can use it
An unfamiliar encoded document uses format A or B, initially with equal probabilities. Two supplied decoders are available: the matching decoder gives a usable output worth 10, and the other gives 0. The model permits one final decoder application. An optional diagnostic costs 1 on the same gain scale and leaves the format unchanged.
The diagnostic returns + with probability 0.8 in format A and 0.2 in format B; the complementary probabilities give −. Its result arrives before decoder selection. The supplied observation model and A.3.3.PI’s conditioning give:
| Information before decoder choice | Probability of format A | Best decoder | Expected final gain |
|---|---|---|---|
| No diagnostic | 0.5 | A or B | 5 |
| Diagnostic + | 0.8 | A | 8 |
| Diagnostic − | 0.2 | B | 8 |
The decision state can be (stage, p), where p is the current probability of A. At the final stage,
V_2(p) = max(10*p, 10*(1-p)).
Each diagnostic result has probability 0.5 under the initial model. Thus
skip diagnostic: V_2(0.5) = 5
use diagnostic: -1 + 0.5*V_2(0.8) + 0.5*V_2(0.2) = 7.
The continuation’s choice changes with the observation. Fixing one decoder in advance gives 5 before the diagnostic cost, or 4 after it. This is why the extra information has value here.
Now the diagnostic result is delayed until after the final decoder choice. The decoder choice can no longer depend on that result. Using the diagnostic then has net expected gain 4, so skipping it gives the better value 5. This repair changes the information timing rather than the arithmetic of conditioning.
A different changed condition also reverses the comparison: with a timely but weaker symmetric diagnostic that is correct with probability 0.55, using its result gives -1+10*0.55=4.5. Skipping still gives 5 and is preferable.
The decoder and diagnostic behavior are premises of this example. A real use requires the applicable processing and observation methods to supply those consequences.