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 agregation 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 demonstratably 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, I assert that my overall
probability of guessing correctly is an INTEGER.  My reasoning is 
that 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 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 agregation 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 
agregate.  I would suggest that, since the "probability of an 
unweighted agregate" 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
probablility 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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