Can the Kolmogorov definition of the conditional probability be proven

[cdux]

It seems weird that such a relatively complex concept is simply given as a definition in most textbooks and then dismissed for further explanation other than using it intact or as a basis for further proofs.

 

[Stephen Tashi]

In mathematics, definitions are not proven. A definition has the form: "[Undefined statement] will be defined to mean [Defined statement]". It's simply a matter of human culture whether a definition is widely accepted. You can't prove a definition is true. It has the same status as an assumption.

If we have two definitions, we can investigate whether the two things that are defined are equivalent. If we have a definition, we can investigate whether the thing that is defined actually exists in the mathematical sense, by showing an example of it or trying to prove no example can exist. If you have something along those lines in mind, you should explain your question.

 

[cdux]

I found one,

https://en.wikipedia.org/wiki/Conditional_probability#Formal_derivation

 

[cdux]

In mathematics, definitions are not proven. A definition has the form: "[Undefined statement] will be defined to mean [Defined statement]". It's simply a matter of human culture whether a definition is widely accepted. You can't prove a definition is true. It has the same status as an assumption.

If we have two definitions, we can investigate whether the two things that are defined are equivalent. If we have a definition, we can investigate whether the thing that is defined actually exists in the mathematical sense, by showing an example of it or trying to prove no example can exist. If you have something along those lines in mind, you should explain your question.

The quirk is that in this particular case the concept is very intuitive [edit: not the derivation or the result] and it stinks from miles that it could be proven, and not surprisingly I found a derivation of it from more basic concepts on the wikipedia article linked in here.

 

[Stephen Tashi]

I found one,

https://en.wikipedia.org/wiki/Conditional_probability#Formal_derivation

That's an example of proving two definitions are equivalent. Supposedly it shows "the formal definition" is equivalent to the Kolmogorov definition.

 

[cdux]

The problem is that the formal definition is perfectly intuitive and a mathematical description that does not satisfy it would be of little use. e.g. defining P(A|B) = 4 makes no sense. By having the conceptual description and then moving on to derive it seems to be the natural approach of introducing the concept, instead of defining the formula, not deriving it and then moving on using it, because while the concept is intuitive, the formula is not when presented out of the blue.

edit: Especially when it is shown it can be derived. It's almost as though Kolmogorov proved it but he was trolling, or he proved it but wasn't sure, or his brain was weird and he could see the formula but not where it came from. It could be also simple intuition from "when approaching P(AB)=P(A)P(B) when they are independent it seems to satisfy it", though if one reaches that point surely he would have known there's a proof nearby.

 

[Stephen Tashi]

I agree that intuitive treatments are comforting. However they aren't a sound basis for proofs. That article's statement of "the formal definition" is imprecise. It isn't the formal definition used in rigorous proofs. The article's version of the Kolmogorov definition doesn't have enough detail to be useful in the modern approach to probability via measure theory, which is an approach attributed to Kolmogorov.

Primitive intuitions about probabiity theory assume that if we have a set of "outcomes" then one can pick an arbitrary sets of outcomes A and B have not trouble speaking of "the probability of A", "the probability of B", "the probability of A given B". This works reasonably well for discrete random variables. But when you have a continuous valued random variable, such as a gaussian random variable, it isn't clear that the probability of an arbitrary set A of outcomes exists. (In the usual axiomatic treatment of proability in that situation, not all sets A can be assigned a probability.)

There are also interesting features of the functions involved in probability. For example, P(A|B) can't be 4, but a conditional probability density function might take on the value 4.

 

[cdux]

I'm not an expert when it goes to formulation of rigorous scientific definitions, but if the problem is that wikipedia's article's version, surely one could formulate a better one that satisfies what one might need, or even satisfy all cases that might needed in the long term, though I wouldn't know if that's ever possible for any mathematical concept (to have 'eternal' coverage).

 

[ssd]

No, the definition cannot be derived or proven. Apparently, one can make some assumption and can derive the said result in consideration with other axioms. But, "making some assumption" implies that the very assumption is to be proven again, which cannot be done. So, in reality, the definition cannot be proven.

 

[cdux]

I think it's pretty straightforward to derive the mathematical formula that describes that definition (e.g. the one referenced there as 'formal definition') and hence one could say "We will now prove the mathematical formula of the formal definition". Are we playing with words here? Because if I didn't call it a definition but I said "I presume that the equation that describes that thing I have in my mind which is <insert that definition here> is this equation and I can prove it", wouldn't it be a proof of that?

Wouldn't that be a form of a 'Conjecture' that went on to be proven?

 

[ssd]

Sorry for the fact that I missed the Wiki's reference. But note the fact that it says "it is reasonable to assume" in formal derivation (in compliance with my previous statement). Matters being somewhat circular here.

 

[cdux]

I start suspecting that what Kolmogorov did was to omit a linguistic definition that he had in mind which was then on derived by him to that simple equation because he thought of it not as generally applicable as he'd like, so he went on with simply providing the formula itself as a definition. I seriously doubt that he didn't have some sort of linguistic [edit: or at least conceptual] interpretation of the concept at hand before deriving the actual equation with the help of tools himself had invented but he probably thought providing it to the world in that manner would not be as theoretically robust.

Perhaps he wasn't a 'words' man as much as a 'numbers' man to say it simply. If he thought that a derivation of the type that is on wikipedia was too obvious, then providing the formula only, was for him almost interchangeable with providing the wording of a linguistic definition.

 

[Stephen Tashi]

I start suspecting that what Kolmogorov did was to omit a linguistic definition that he had in mind which was then on derived by him to that simple equation because he thought of it not as generally applicable as he'd like, so he went on with simply providing the formula itself as a definition..

The wikipedia article does not describe Kolmorgorov's approach to probability theory. It merely uses the terminology "Komogorov definition". Your suspicions about his method are too simplistic. If you study continuous random variables and measure theory, you can understand the Kolmogorov approach.

 

[cdux]

Why don't you enlighten us?

 

[ssd]

The wikipedia article does not describe Kolmorgorov's approach to probability theory. It merely uses the terminology "Komogorov definition". Your suspicions about his method are too simplistic. If you study continuous random variables and measure theory, you can understand the Kolmogorov approach.

Could not agree more.

Kolmogorov brought conditional probability under well defined measure.

 

[ssd]

I start suspecting that what Kolmogorov did was to omit a linguistic definition that he had in mind which was then on derived by him to that simple equation because he thought of it not as generally applicable as he'd like, so he went on with simply providing the formula itself as a definition. ... he wasn't a 'words' man as much as a 'numbers' man to say it simply.

Agree with you partially.

A linguistic definition in case of mathematics is often non rigorous and symbolic definition is rather concrete with all its pros and cons...I may say unambiguous. Mathematicians ultimately has to work with the symbolic definitions.

It appears (to me) that Kolmogorov intentionally avoided a linguistic definition and the reasons for me are:

1/ He was a man of great imagination. It is not that he could not give the linguistic, but he (certainly) knew that the concept of conditional probability will not be as before and also about the ambiguity to arise. He avoided.

2/ The symbolic definition was sufficient and satisfactory to him at that point of time (early 1930's) as foundation of his work.

3/ He was (naturally) engrossed in developing the main building blocks initially. When the main blocks turned out to be overwhelmingly successful, he, either did not want to reinterpret (disturb) his symbolic concept (because the goal was achieved) or was satisfied with the new concept of conditional probability.

PS. The debate in modern days is still under progress around the point that whether priors are basic or the posteriors are basic.