Effects of Unknown Dependency Between Two Measurements

This investigates example 8-13 in chapter 8, Statistical Decision Theory, extended to add a second measurement with the same characteristics as the first measurement. In so doing, the dependency relationship between the two measurements is considered undefined.

Observations 

Without knowing the dependency relationship between the measurements, expected loss will not be knowable to the precision of a number, but rather only knowable to within an interval [l,h]. For example, the expected loss for decision plan d80 is in [1.25,1.7]. The midpoint (arithmetic or geometric) does not seem to have any special relationship to reality. It does not, for example, lead to the same loss estimate as assuming independence. 

The maximum difference between the actual expected loss and the expected loss calculated with an incorrect assumption about the dependency relationship between the measurements is h-l.

If the width of the interval bounding the range of losses that would occur in a given measurement situation, over the set of possible dependency relationships, is twice as wide for problem X as for problem Y, then presumably the expectation for the difference between an estimated loss given the assumption of any particular dependency relationship, and the more correct loss that would be calculated if we knew the correct dependency relationship, would be twice as great for problem X as for Y. Thus the risk would be higher for X, with all the (probably qualitative) inferential implications that leads to.

An incorrect assumption of dependency between measurements can lead to an incorrect estimate of loss for a given decision plan. No big surprise there. For example, decision d80 has the lowest loss, 1.25, for one dependency relationship, but a significantly higher loss of 1.7 for another dependency relationship.

An incorrect assumption of dependency between measurements can lead to mistakenly choosing a decision that, while optimal for the assumed dependency relationship, is not optimal for the actual dependency relationship. For example, decision d80 has the lowest loss, 1.25, for one dependency relationship, but if the actual dependency relationship was the one for which its loss is 1.7, another decision plan would be better because there are others for which the loss is only 1.55 for that dependency relationship. These cost differences are quantifiable. For example, mistakenly choosing decision plan d80 could mean an expected loss of 1.7 instead of the 1.55 which should have been achieved.  

The value of knowing the dependency relationship is in [0,h-l], if the range of losses possible over the set of dependency relationships is [l,h]. One might, for example, be paying for the opportunity to play and interested in bidding the optimal amount.

Results are in decisionAnalysisUnknownDependency.xls.