Lovegrove Mathematicals

"Dedicated to making Likelinesses the entity of prime interest"

Often, there is no need to be
concerned about the value on the RHS, it being enough to note that if g_{1},g_{2} have the same degree and sample size then
L_{S(N)}(g_{1})=L_{S(N)}(g_{2}).

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

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

There is a geometrical way to obtain the same result.

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... 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 (http://www.thefreedictionary.com/eduction") "To infer or work out from given facts". are concerned with the possible occurrences of the determinates p_{1}
... p_{n} under the determinable P.

______

(1) Combination Postulate

In a total of M instances, any proportion, say m_{1}:m_{2}: ... :m_{α}
where m_{1}+m_{2}+...+m_{α} = 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 m_{1}:m_{2}: ... :m_{α} 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

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.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 (http://www.thefreedictionary.com/eduction") "To infer or work out from given facts". are concerned with the possible occurrences of the determinates p

______

(1) Combination Postulate

In a total of M instances, any proportion, say m

(2) Permutation Postulate

Each of the different orders in which a given proportion m

______

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

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'.

- Johnson uses the expression ``prior to any knowledge of the occurences in question''. That knowledge can come from two places: theory and observation, so there must be no knowledge from either source. No knowledge from theory suggests that the underlying set should be S(N); no knowledge from observation suggests that the given histogram should be 0. So, for example, if we consider the tossing of a coin then all we know about the probability-pair (Pr(``H''),Pr(``T'')) is that it is --as all probability-pairs must be-- somewhere on the line segment from (0,1) to (1,0)
- So far as the meaning of the word `likely' is concerned, there are two possible contenders: probability and best-estimate of probability, ie. likeliness. It has to be remembered that we are specifically maintaining the distinction between the two. In the formal wording of the Combination Postulate, Johnson uses the word `likely' but does not actually refer to probabilities. He does use the word `probability', but only outside of that formal wording. This admits the possibility that, when drafting the formal wording, Johnson may have been thinking (albeit at an intuitive level) of a wider concept than `probability' but subsequently interpreted it as meaning specifically probability. Whether or not this was the case must, of course, be a matter of speculation but the condition he states does suggest that he may have been thinking about the expected value of the probability, ie. what happens on average, rather than the probability itself.

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

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

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.