Originally Posted by uppitychick,Nov 6 2005, 10:00 PM

The whole ranking thing is just too complicated.

Assume that you are asked to rank scientific theories, general rules, or distributions, according to their likelihood. You may based your ranking on observations. These can be the outcomes of a roll of a die, of a surgical procedure, and so forth. Let a memory be a vector, counting how many times each case has been observed, and assume that you can rank theories based on each such memory.

Assume further that your rankings satisfy the following combination condition: if theory T is considered more plausible than theory T' given two disjoint sets of observations, then T is considered more plausible than T' also given the union of these sets.

We impose two more conditions, one of continuity (or archimedeanity), and another of diversity. The conclusion is that one has to rank theories according to their likelihood function.

That is: for any such collection of rankings (of theories given any possible memory) satisfying our axioms, there exists a matrix of conditional probabilities p(c|T), denoting the conditional probability of case c given theory T, such that, given any memory, the plausibility ranking agrees with the likelihood function of the theories given the data.

Originally Posted by WestSideBilly,Nov 8 2005, 08:54 AM

It's only complicated if you factor in conditional probabilities and axioms.