Borel–kolmogorov Paradox Puns

Hello everyone

I have recently learned about the existence of the Borel-Kolmogorov paradox which says in essence that the act of computing a conditional probability distribution is ill-defined on continuous probability space (when you condition on an event that has probability 0), and I'm trying to further my understanding of it so that I can be safe from mistakes.

In the discrete world: [;p(x|y) = \frac{p(x,y)}{p(y)} ;]. However, if we use this formula in the continuous world, the paradox manifests itself as [; p(X|Y=y_0) ;] and [;p(X|U=u_0);] being potentially different when the change of variable between Y and U is such that [;Y(u_0)=y_0;].

The [english wikipedia article] (https://en.wikipedia.org/wiki/Borel%E2%80%93Kolmogorov_paradox) offers a good reference with an example.

The reason for the ill-definition of the conditioning event is that you have to specify what limit process takes you to your probability 0 event. Different processes can have the same limit probability 0 event, while producing different conditionals. When you use the formula I wrote above to condition of $Y=y_0$, that apparently corresponds to conditioning on [;Y \in [y_0-\epsilon, y_0+\epsilon];] and letting [;\epsilon \rightarrow 0;]

In order to justify one approximation frame instead of the other, I think you can appeal to the following physicalist intuition: you can't measure values exactly for your random variables. What you measure instead is [;(true Value + \epsilon noise);] where [;\epsilon;] is a small constant, and the noise is Gaussian. If I'm not mistaken, the conditional in the limit [;\epsilon \rightarrow 0;] gives the conventional formula. The presence of this noise justifies one frame of reference being superior to the other. Even if you now operate a change of variable, your limiting sequence will still be defined in the original space.

This idea of an infinitesimal measurement noise is a way to avoid problems that you encounter in the continuous probability domain. Has this been formally explored by some authors ? My only reference on the danger of the continuous world is Jaynes, and he only swears by coming back to the discrete world, which is of course fine, but I'm looking for something different.

Do you have some other interesting insights on Borel-Kolmogorov ?

ps: why does my Tex look all fucky ? I followed the sidebar advice :-(

I did find proof but it was using code as examples I did not understand fully or was too formal in its proof. I need a variant which isn't that formal in the proof.

Basically everything's in the title already (I guess).

https://en.wikipedia.org/wiki/Probability_axioms

I read earlier today, that Kolmogorov's Probability Axioms are some of the most important results in probability.

Can someone please explain why these are so important? What relevance and application do they have?

Thanks

https://imgur.com/a/7ZLkRPI

Suppose we have three variables : x1 (predictor), x2 (predictor) and y (target). We are interested in making a regression model y ~ x1,x2. We have some current data, and we make the regression model (could be any machine learning or statistical model, I just used regression for simplicity). This model is working fine, but now the question of "concept drift" arises : How do we know that this regression model will keep working in the future? What if the underlying process that is generating the data has started to (significantly) change? Perhaps it's time to retrain the model, or to restart completely?

I understand that this is a very abstract concept and that there is no single answer to deal with this. But here is an idea I had : suppose we decided to compare all the marginal and conditional distributions for all combinations of variables, for old vs new data - if these distributions are "statistically similar" (e.g. determined by the Kolmogorov-Smirnov test) - is there some grounds to believe that no major form of concept drift has occurred? I understand this might not be the best approach - converting the observed data into continuous probability distributions (using kernel density estimation) comes with its own statistical risk, these continuous probability distributions might not fit the data well, therefore the results from the Kolmogorov-Smirnov test might be deceptive.

What does everyone think of this approach for studying concept drift?

Thanks

(would it be better to use the Kullback-Leibler Divergence instead of the Kolmogorov-Smirnov test?)

Hi! I’m wondering if anyone has any knowledge about this test/it’s applications.

I’m using it to compare the cumulative distributions of egg production between groups of tardigrades, but I’m not sure how to account for changing numbers of tardigrades in each group.

Ex: 2 tardigrades in 1 sample have died, leaving 8. These 8 produce 10 eggs, but a different group of 10 also produces 10 eggs.

To my understanding, this test only compares the eggs produced, ie comparing 10 to 10, without considering the differing size of the groups that produced these eggs. How should I account for these changing group sizes?

Thanks in advance, I’d be happy to explain further if this was unclear!

Does anyone know about this? If so then I want to know why only kinetic energy’s Fourier spectrum is plotted? Isn’t there any other form of energy contributing to the total energy?

I was reading this article here: https://observablehq.com/@nstrayer/exploring-the-universal-approximation-theorem

"The universal approximation theorem says that we can approximate any borel-measurable function with a neural network with a single hidden layer. Provided that hidden layer can go to infinite length."

Can someone please explain to me in simple terms, what is a borel set?

I tried reading the wikipedia page: https://en.m.wikipedia.org/wiki/Borel_set

But this didn't make a lot of sense.

Can someone please help me understand what is a Borel Set?

Thanks

In the first or the second class in a probability course at university, they will always give you the 3 axioms of Kolmogorov:

1. P(Ω) = 1
2. P(A) >= 0
3. The probability of the union of disjoint events is the sum of the probabilities of those events

But they didn't teach me where do this 3 come from. I mean, how did Kolmogorov come up with that the 3 statements here were the probability axioms you could use for building the whole probability theory?

And, about the 3 axiom, why is that an axiom and how would you show that it holds for any set of events if it wasn't considered an axiom?

"Em um documento obtido pelo portal, o juiz da 4ª Vara Cível do Rio de Janeiro, Marco Antonio Cavalcanti de Souza, disse: “Sem adentrar no mérito, entendo que, atualmente, diante de crescentes quantidades de casos de feminicídio, não se pode admitir qualquer utilização de meios jurídicos para que o suposto ofensor possa desqualificar os relatos de sua ex-companheira”. Pode isso? E sério o chegamos ao ponto em que o homem não pode se defender na porra do tribunal, Mano mas que bosta se continuar assim irá acontecer danos inreversiveis para o sexo masculino.

https://en.wikipedia.org/wiki/Probability_axioms

I read earlier today, that Kolmogorov's Probability Axioms are some of the most important results in probability.

Can someone please explain why these are so important? What relevance and application do they have?

Thanks