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I remember a proof that Bayesian probability theory is the only valid method for representing beliefs, it went something like

  1. we represent belief by some non-negative function over some domain of outcomes
  2. beliefs are sub-additive
  3. ...

Therefore, Bayesian probability theory is the only valid approach for representing beliefs.

The idea is that under very basic, and general, assumptions for what constitutes a "belief function", you end up modeling "belief" with Bayesian probabilities.

I've forgotten where I've seen it.

Does anyone know this proof? or a reference to the original?

Edit So far the best lead I've found is that it is presented in:

Savage, L. J. (1954). The Foundation of Statistics, 2nd edn, Dover, New York.

(which I don't have a copy of)

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  • $\begingroup$ I would close this question as off-topic because it's better suited for the philosophy SE, but apparently I can't. Nonetheless, I encourage you to look at previous posts on Bayesian probability there and see how you might migrate your question for that audience. $\endgroup$
    – AdamO
    Commented Mar 25, 2014 at 19:57
  • $\begingroup$ The question of whether this post is on topic is currently under discussion at meta.stats.stackexchange.com/questions/2005. I would suggest keeping it open here, because it is on topic and might generate some good answers, but in the event nobody responds within a few days we can easily migrate it to the philosophy site. $\endgroup$
    – whuber ♦
    Commented Mar 25, 2014 at 20:19

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So far I've seen two threads along these lines:

One of the earlier attempts is Cox's Theorem (Cox, R. T. (1946). "Probability, Frequency and Reasonable Expectation". American Journal of Physics 14: 1–10), which essentially assumes Bayes theorem, and then derives the features of the resulting belief functions, and finds them to be the laws of probability. Later, this approach was more fully explicated in E. T. Jaynes Probability Theory: The Logic of Science (the first few chapters are online), and summarized on Wikipedia.

Another thread comes of of Savage's formulation of decision theory (Savage, L. J. (1954). The Foundation of Statistics, 2nd edn, Dover, New York.). Here the key assumption is that one can rank-order linear combinations of different outcomes/decisions. This allows one to impose an additive structure on the utility function, which is then conceptually factored into "value" and "belief" parts; the belief part behaving as per probabilities. One problem is that the factoring is not unique, however, for the purposes of constructing a model for belief, the utility function is, essentially, just a 0-1 loss function. Thus it drops out of the representation and you are left with probabilities as being the representation of belief. (I'm basing this discussion on Edi Karni _Savages' Subjective Expected Utility Model, JHU Tech Report(?), 2005)

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