Lovegrove Mathematicals


"Dedicated to making Likelinesses the entity of prime interest"

Combination Theorem

Combination Theorem


Often, there is no need to be concerned about the value on the RHS, it being enough to note that if g1,g2 have the same degree and sample size then LS(N)(g1)=LS(N)(g2).

This is not to say that the RHS is not important. It is actually very important since it follows that
Order of Sample Space

There is a geometrical way to obtain the same result.

Informally, the Combination Theorem applies when we "know nothing" about the generating distribution.

Using S(N) as the underlying set means that we have no theoretical reason for eliminating any distribution as a possibility. Having the zero histogram as the given histogram means that we have no information from observations.

This "know nothing" property makes the Combination Theorem very useful when we are trying to develop general principles.

The Combination Theorem is here named after the Combination Postulate, which was proposed by William Ernest Johnson but not proved by him. After trying for some time, Johnson eventually gave up relying on, or trying to prove, this Postulate.

Johnson was a late 19th-early 20th century logician, based in Cambridge, working on probability theory and economics. His work is important in the history of the development of probability theory since it was linked to, and a close forerunner of, de Finetti's work on exchangeability.

At the time of his death in 1931, Johnson was working on a 4-volume work called Logic, the first three volumes of which were published posthumously; the fourth volume was not completed. In Volume 3 he wrote:-

... the calculus of probability does not enable us to infer any probability-value unless we have some probabilities or probability relations given.

The following two postulates in the Theory of Eduction This is not a mistake. The word really is "Eduction"; it is not meant to be "Education". It means (") "To infer or work out from given facts". are concerned with the possible occurrences of the determinates p1 ... pn under the determinable P.

(1) Combination Postulate

In a total of M instances, any proportion, say m1:m2: ... :mα where m1+m2+...+mα = M, is as likely as any other, prior to any knowledge of the occurrences in question.

(2) Permutation Postulate

Each of the different orders in which a given proportion m1:m2: ... :mα for M instances may be presented is as likely as any other, whatever may have been the previously known orders.

In what follows certitude will be represented by unity. By (1), the probability of any one proportion in M instances  equal likelinesses

I have here partitioned off the formal mathematical statements of his two postulates by horizontal lines. The paragraphs before and after them are informal commentary. I have also written the words `probability' and `probabilities', wherever they occur, in red. I have done this to emphasize that Johnson did not use the words 'probability' and 'probabilities' in the formal mathematics, only in the informal commentary.

Combination Postulate

Imagine rolling a die. In any one roll, there are six possibilities, or `determinates', namely "1", "2", ... , "6"; so α =6. If we were to roll the die 10 times then there would be 10 `instances' of those determinates; that is M=10. Say the number of rolls of each face were (1, 3, 0, 2, 2, 2) respectively: these are the values of the mi. Of course, 1+3+0+2+2+2=10: that is, we have an ordered 6-tuple of non-negative integers summing to 10: this is an ordered 6-partition of 10.

What is confusing to the modern eye is Johnson's use of the word `proportion' to refer to something which we would not usually think of as a proportion. He is using it to refer to an ordered 6-tuple such as  (1, 3, 0, 2, 2, 2), ie. what we are here calling an integram. The Combination Postulate, when it says that any proportion is as likely as any other, is saying that any integram is as likely as any other of the same sample size.

There are two questions remaining about Johnson's wording, concerning the circumstances under which the integrams are equally likely, and the meaning of the word `likely'.
So could it be that Johnson's wording of the Combination Postulate was correct but that his stated interpretation of it in terms of probabilities, rather than expected values of probabilities, was not? This would certainly cause him difficulties, as we know happened.

The Combination Theorem answers this with "YES".
The proof of the Combination Theorem can be found in Fundamentals of Likelinesses.

Permutation Postulate

Any integram can be considered as an integram of observations, and those observations -which would be made as a sequence of observations- could arise in several orders. The integram (2,3), for example, could have arisen as the sequence "1","1","2","2","2", or as the sequence "2","2","1","2","1", or as the sequence, "2","2","2","1","1", etc. These are the orders to which Johnson refers.

What Johnson is saying in the Permutation Postulate, is that we pay no attention to the order in which observations are made, only to the final total. The order in which the observations leading to (2,3) occurred is not important, only the fact that the final histogram is (2,3).

There are exceptions to this (for example, if less weight is attached to older observations), but generally speaking this is true for both likelinesses generally -regardless of the underling set- or for specifically probabilities -with singleton underlying sets.

Since the Permutation Postulate applies regardless of whether or not the underlying set is singleton, Johnson would not have encountered any difficulties by considering only probabilities - as he appears to have done with the Combination Postulate.

This postulate is well-known as the introduction of the concept of exchangability