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276 DS perspective on statistical inferencelonger and shorter intervals. The issue here is a version of the “multiplicity”problem of applied inferential statistics.Several modified DS-ECP strategies come to mind. One idea is to“strengthen” the broad assumption that allows all continuous populationCDFs, by imposing greater smoothness, for example by limiting considerationto population distributions with convex log probability density function.If the culprit is that invariance over all monotone continuous transforms ofthe scale of the observed variable is too much, then maybe back off to justlinear invariance as implied by a convexity assumption. Alternatively, if it isdesired to retain invariance over all monotone transforms, then the auxiliariescan be “weakened” by requiring “weakened” auxiliary “don’t know” termsto apply across ranges of intervals between data points. The result would bebets with increased “don’t know” that could help protect the broker againstbankruptcy.24.6 Open areas for researchMany opportunities exist for both modifying state space structures throughalternative choices that delete some variables and add others. The robustnessof simpler models can be studied when only weak or nonexistent personalprobability restrictions can be supported by evidence concerning the effects ofadditional model complexity. Many DS models that mimic standard multiparametermodels can be subjected to strengthening or weakening modifications.In particular, DS methods are easily extended to discount for cherry-pickingamong multiple inferences. There is no space here to survey a broad rangeof stochastic systems and related DS models that can be re-expressed andmodified in DS terms.DS versions of decision theory merit systematic development and study.Decision analysis assumes a menu of possible actions each of which is associatedwith a real-valued utility function defined over the state space structure(SSS). Given an evidence-based posterior mass distribution over the SSS, eachpossible action has an associated lower expectation and upper expectation definedin an obvious way. The lower expectation is interpreted as “your” guaranteedexpected returns from choosing alternative actions, so is a reasonablecriterion for “optimal” decision-making.In the case of simple bets, two or more players compete for a defined prizewith their own DS mass functions and with the same utility function on thesame SSS. Here, Borel’s celebrated observation applies:“It has been said that, given two bettors, there is always one thiefand one imbecile; that is true in certain cases when one of the twobettors is much better informed than the other, and knows it; but it