Choice Without Context?

Suppose someone picks two distinct real numbers A and B. We know nothing about the distribution from which they were picked - they can be any finite real numbers. The person tells us one of the numbers (deciding at random whether to tell us A or B), and we have to guess whether this is the larger or smaller of the two numbers. Does there exist a strategy that assures of us of a better than 50% chance of guessing correctly? The standard answer is "yes", based on the strategy of randomly choosing a number from, say, a normal distribution, and then guessing as if it were the unknown number. Intuitively it certainly seems that this would give the correct answer with probability greater than 1/2. However, this seemingly obvious answer rests on at least one undefined notion plus a couple of questionable principles of inference.

Let U be the set of allowable (and well-defined) contexts for an event x, and suppose that Pr{x|u} satisfies condition C for every u in U. It's tempting to think that Pr{x|U} must also satisfy condition C, where Pr{x|U} is taken to signify the "overall probability" mentioned in the above quote. However, the set U of allowable contexts is not a member of itself. We could define an enlarged set U' that includes a context consisting of an aggregation of all the elements of U, but that would require us to specify a weighting of the elements of U. Lacking that, there is no well-defined "overall context" for x, and without a well-defined context for x we need to be extremely cautious about any assertions characterizing the "probability of x".

It might be argued that, although the overall probability of x is underspecified, we are still entitled to infer that it shares every property possessed by every allowable context. However, unrestricted inferences of this type are demonstrably false. To illustrate, suppose you have selected a number from the set {0,1}, and the experiment is for me to guess which of those two numbers you chose. Given that my strategy is to always guess 0, one might claim that my overall probability of guessing correctly is an integer, because there are only two allowable contexts for the experiment: either you selected 0 or you selected 1. In either case my probability of guessing correctly is an integer, so (by this reasoning) my "overall probability" must be an integer. This fallacy shows that, at the very least, we have to be careful about what types of conditions we "carry over" from the set of allowable contexts to the "overall context". Now it might be argued that the condition {"is an integer"} is somewhat pathological, and that for more typical conditions such as {"is greater than 0.5"} we can safely extrapolate from the individual contexts to the "overall context". However, the lack of a well-defined "overall context" makes even simple bounding conditions questionable.

To illustrate, suppose I have an experiment with infinitely many possible well-defined contexts. No weighting of the possible contexts is specified, but for each context u we have Pr{x|u} > 0.5. Can we legitimately assert that the "overall probability" exceeds 0.5? I would argue that the notion of "overall probability" is so ill-defined in this situation that we cannot say anything meaningful about "it". We can, however, consider a closely related concept that is well defined: the least possible value of Pr{x|u} for any u in U.

Unfortunately, it's entirely possible that the set of values Pr{x|u} does not possess a least value (as in the particular situation posed at the beginning of this article). In such a case, the lowest possible probability (which is a well-defined concept) has no well-defined value. The closest thing to a definite limiting value would be the greatest lower bound for the set S = { Pr{x|u}, u in U }, which of course is 0.5.

The claim that the "overall probability" of guessing correctly exceeds 0.5 amounts to a statement that the set S does not contain 0.5, even though for any e>0 the set S contains an element q such that q − 0.5 < e. This brings us to the crucial point of interpretation: In the same casual spirit in which we are asked to regard U as a context, i.e., as a member of an enlarged set of contexts U', is it not appropriate to regard the greatest lower bound of S as an element of the correspondingly enlarged set of probabilities S'? In passing over from the set of individual contexts u_i to the vague "overall context" consisting of an unweighted aggregation of infinitely many contexts, it isn't self-evident (to me) whether the boundary should or should not be included in the range of possible probabilities for the aggregate. I would suggest that, since the "probability of an unweighted aggregate" is not a previously well-defined concept, we are free to either include or exclude the boundary in our definition of the range.

The original problem stipulated that we know "nothing" about the distribution from which the two real numbers A and B were chosen (although it takes for granted that they must have been chosen from some distribution). Now, to say we know nothing about the distribution is not exactly true, because we know (for example) that the distribution ranges over the real numbers, and the total probability under the distribution from negative to positive infinity is 1. More significantly, we know the distribution is not uniform over the entire set of real numbers, because no such distribution exists. The usual way of stating the question is to say the numbers A and B were chosen "randomly", suggesting that any real numbers are equally likely, but this is misleading precisely because there does not exist a uniform distribution over the reals.

The non-existence of such a distribution has some interesting physical implications. For example, in quantum mechanics the product of the uncertainty in a dynamic variable times the uncertainty in its conjugate momentum cannot be less than a certain positive constant, and this implies that if the uncertainty in one of those variables is zero, the value of the conjugate variable is completely indeterminate, which is to say, all values are equally probable. In "The Principles of Quantum Mechanics" Paul Dirac observed that

It is evident physically that a state for which all values of q [position] are equally probable, or one for which all values of p [momentum] are equally probable, cannot be attained in practice, in the first case because of limitations in size and in the second because of limitations in energy. Thus an eigenstate of q or an eigenstate of p cannot be attained in practice.

Arguably the reason for the inability to achieve an eigenstate is even more fundamental than practical limitations of size or energy, because even from a purely mathematical-logical standpoint there does not exist a distribution for which all values of p (or of q) are equally probable. It follows, in this same sense, that eigenstates do not exist.

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