[extropy-chat] what is probability?

[Ben Goertzel]

Hi,

> 1) The Logical Theory, in which probability is defined as a degree of
> rational belief (Keynes).
>
> 2) The Subjective Theory, in which probability is a degree of belief of a
> particular individual (Ramsey, De Finetti).

The most consistent interpretational approach, I believe, is a fusion
of the Subjective Theory and Logical Theory as enabled by Cox's
Theorem.

I.e., a probability is a crude way of encapsulating a degree of belief
of a particular individual.  And, if an individual is completely
rational, then their degrees of belief will completely obey the laws
of probability.  If an individual is partially rational, then their
degrees of belief will partially obey the laws of probability.

For instance, no highly resource constrained mind is going to be able
to fully obey the third assumption of Cox's Theorem,

http://en.wikipedia.org/wiki/Cox's_theorem

thus perfect probabilism is only for unrealistically resource-enabled
minds like Hutter's AIXItl.

> Consider a frequent event (E), such as 'Rain in the Amazon Rain Forest'.
>
> Which statement is most true?
>
> A) E is frequent because it is probable.
> B) E is probable because it is frequent.

E is rationally estimated as probable in the future, because it has
been observed as frequent in the past.

A related point is that single-number probabilities are not
necessarily the best way to describe a system's degree of belief.
Keynes suggested interval probabilities, and in the Novamente AI
system we work with what we now call "indefinite probabilities",
intervals [L,U] with the interpretation

"I estimate that, after N more observations, my probability estimate
of the event E will lie in the interval [L,U] with probability b."

This is a more sophisticated approach than Keynes' interval
probabilities or Walley's imprecise probabilities but with a similar
underlying philosophy.

-- Ben