Jeffreys’ Probability

Harold Jeffreys The Theory of Probability 3rd Ed. Clarendon Press, Oxford, 1961.

Jeffreys’ classic ‘objective Bayesian probability theory’, striving for objective foundations to what is now common practice.

Preface to 3rd Edition

Jeffreys notes Keynes’ alternative theory of partially ordered probabilities, and claims that Keynes retracted the need for this in his biographical review of Ramsey.

Fundamental Notions

Jeffreys (pgs 4/5) thinks it reasonable to suppose that the truth is the simplest possible explanation:

I say … that the simplest law is chosen because it is most likely to give correct predictions; that the choice is based on reasonable degrees of belief  … . … It is … sometimes said that trust in the simplest law is a peculiarity of human psychology .. . … a general statement that something accepted by the bulk of mankind is intrinsically nonsense requires much more to support it than mere declaration.

He also declares Russell & Whitehead’s principia to be false, at least in so far as it conflicts with his own views.

Jeffreys considers a body freely falling under gravity, noting that from a number of data points one could ‘join the dots’ in many ways, so there is no unique solution to the equation of motion. He notes that one typically chooses the simplest.

Comments

Jeffreys is sometimes regarded as founding an objective Bayesian probability theory, and does seem to have striven for objective (and logical) solutions. But more generally, the dependence on simplicity is questionable, as is the existence of objective probabilities outside ‘small world’ problems.

As Keynes noted, it does often to be the case that the simplest solution is somehow the best, perhaps because of some principle of uniformity in nature. But simply observing, as Jeffreys does, that such a principle often seems to be obeyed and very often seems to be assumed, does not establish it. It may be that, as Whitehead, Keynes and Russell suppose, more complex situations are possible.

With regard to Jeffreys’ example of a falling body, one may have two problems:

  • It is not always clear which is the simplest solution.
  • It may be that the body is falling towards a planet, in which case its motion may suddenly stop.

Thus Jeffreys adds to Keynes the idea that there are universal rules by which one can always assign ‘objective’ probabilities, without providing them. Jeffreys’ notion, to extrapolate from past experience unless and until one has contradictory new data, is surely pragmatic. But a better approach may be to look out for alternative theories and diagnostic data before one has a such a concrete problem. In this it seems helpful to be able to distinguish between probabilities that are truly logical /’ objective’ and those that are merely pragmatic / conventional.

See Also

My notes on uncertainty, and on other probability classics.

Dave Marsay

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