Classical definition of probability & kolmogorovs axioms

[macca1994]

I've seen in some probability theory books that the classical definition of probability is a probability measure, it seems fairly trivial but what is the proof for this? Wikipedia gives a very brief one using cardinality of sets. Is there any other way?

 

[micromass]

There is no proof. It's a definition. Definitions never have proofs. The only thing you can ask is why this definition is the right one and why it encapsulates our naive understanding of probability. To answer this question, you need to work out some examples of probability spaces. For example, take two dice and throw them. Describe the probability space and see if the theory matches the experiment.

 

[ssd]

One is Kolmogorov's axiomatic approach, as you mentioned in the title. Another is Baye's conditional approach...which is a different measure than the classical one.

 

[macca1994]

oh okay, so you prove that it obeys the axioms by kolmogorov?

 

[ssd]

oh okay, so you prove that it obeys the axioms by kolmogorov?

Which one? Classical defn.? The answer is no, these two are different measures.

 

[dkotschessaa]

oh okay, so you prove that it obeys the axioms by kolmogorov?

Using fairly basic set theory principals and Kolmogorov's 3 basic axioms you can prove/create more postulates (much like the Euclidian approach to geometry). Then from there you can show how and why things in classical probability work the way they do. You can also state Bayes laws in terms of Kolomogorov axioms.

-Dave K

 

[Stephen Tashi]

what is the proof for this? Wikipedia gives a very brief one using cardinality of sets.

I doubt it. As micromass said, definitions don't have proofs. A definition is not a theorem. Perhaps what you saw in the Wikipedia was a demonstration that the cardinality function on finite sets statisfies the definition of a measure. Such a demonstration is an example of a measure, not a proof of the definition of a measure. (When a mathematical definition for something is invented it is reassuring to demonstrate that at least one example of the thing exists. Such a demonstration does not make the definition "true". It only indicates that it is not futile to study the thing that is defined.)

 

[D H]

One is Kolmogorov's axiomatic approach, as you mentioned in the title. Another is Baye's conditional approach...which is a different measure than the classical one.

Bayes' (not Baye's) theorem is consistent with Kolmogorov's axiomatization of probability. It derives from Kolmogorov's axioms.

Yes, there is a dispute between frequentists and Bayesianists over the meaning and validity of prior probabilities, but that's a debate over interpretations of probability, not over fundamentals.

 

[ssd]

Bayes' (not Baye's) ...

Right. Thanks.

What I really intended to mean is, through Bayes' approach we (first) had the taste of having a different probability measure than the classical approach. More than a century later we again had another probability measure due to Kolmogorov, differing from the classical one. Historically, Bayes' stood alone for many years fighting(!) with the classical.