After 2.9 in the text: “Furthermore, the function F(x, y) must be continuous; for otherwise an arbitrarily small increase in one of the plausibilities on the right-hand side of (2-1) could result in a large increase in AB|C.”
Is there some particular reason that’s an unacceptable outcome, or is it just generally undesirable?
(I suppose we might be in trouble later if it weren’t necessarily continuous, since it wouldn’t be necessarily differentiable (although he waves this off in a footnote with reference to some other papers and proofs) so this seems like an important statement.)
Jaynes mentions a “convenient” continuity assumption following 1.28, and uses it following 1.37. As you point out, the comments following 2.13 seem to indicate something of why Jaynes believes this assumption to be only convenient but not necessary. The comments of talyo just below suggest that Jaynes was wrong—we need something approximating continuity.
But continuity is not difficult to justify. We need only assume that we can flip a coin an arbitrarily large number of times and recall the results. Hmmm. Recall an arbitrarily large (unbounded) quantity of information? Maybe it is not so easy to justify...
After 2.9 in the text: “Furthermore, the function F(x, y) must be continuous; for otherwise an arbitrarily small increase in one of the plausibilities on the right-hand side of (2-1) could result in a large increase in AB|C.”
Is there some particular reason that’s an unacceptable outcome, or is it just generally undesirable?
(I suppose we might be in trouble later if it weren’t necessarily continuous, since it wouldn’t be necessarily differentiable (although he waves this off in a footnote with reference to some other papers and proofs) so this seems like an important statement.)
Jaynes mentions a “convenient” continuity assumption following 1.28, and uses it following 1.37. As you point out, the comments following 2.13 seem to indicate something of why Jaynes believes this assumption to be only convenient but not necessary. The comments of talyo just below suggest that Jaynes was wrong—we need something approximating continuity.
But continuity is not difficult to justify. We need only assume that we can flip a coin an arbitrarily large number of times and recall the results. Hmmm. Recall an arbitrarily large (unbounded) quantity of information? Maybe it is not so easy to justify...