TommyTorty10

february

@TommyTorty10

  • they/them

likes breaking electronics, fiddling with old computers, and making music
https://tommytorty10.gitlab.io/pages/

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TommyTorty10
@TommyTorty10

I'm realizing that I think of stats nearly entirely geometrically, while it's often taught nearly entirely algebraically.

Time for some mad mathematical ramblings I came up with late last night and have not edited or reflected on since.

Conditional probabilities are just the intersect of some probabilities scaled down by the condition. Can I also scale down a statement like (A union B)? What would that mean? P(AUB | B) = ? This would be the intersect of AUB and B scaled down by B. That is to say B scaled down by B, which would be 1. I think I've imbibed enough statistics tonight. Let's find out if statistical poisoning exists. what happens if I scale something up? what is A scaled up by B? I know that a conditional can be scaled up to get an intersection, so what if I substitute nothing as the condition for A?

Then our condition is equal to the intersection of A and 0 scaled down by 0. Aaaaaand we're dividing by 0. so I can scale between sorta 2 different spaces. Why? Can I ever get 3 spaces? What if I have a condition based on a condition? P(A | (B|C)) = ?

P(B|C) is the intersection of B and C scaled down by C. P(A | (B|C)) is the intersection of P(B|C) and A scaled down by P(B|C) I've got 2 scale operations in what sorta feels like 1 direction. I scaled from the space that B and C are in, into a condition, and then used them to scale into another condition.

So what about P((B|C) | A)? Then, I, in effect, scale B twice. I now have the intersect of (B|C) scaled down by A. So I have the intersect of A and (the intersect of B and C scaled down by C) scaled down by A. So I can scale as much as I want if I have more variables to create conditions

So I can use intersections and unions to subtract and add areas, and given enough variables, I can use conditions to scale areas. pours the statistics into a bong How can I get 3 dimensional? It's just a collection of points, so there's nothing to stop us from envisioning our current model a 3D one instead of a 2D one. Or is there? 3 dimensions really only has 1 difference, translation in another dimension. How can I translate shit? Effectively, I'm changing the intersection between things and then redefining values based on that intersection. This feels like dangerous territory. If A and B intersect, and I change that intersection, I just push A and B farther apart. If C also intersects with A and B, they all separate relative to each other. We can also view this as some scaling. Changing the intersection requires us to scale up our area to produce the values of the variables. Each scaling operation requires knowing a particular intersection. If I change the intersection between (B|C) and A, but not the intersection between B and C, then I effectively translate A away from B and C. Now how do I lift off into the 3rd dimension? I need a point out there in 3D space to scale down towards. And we're back to just having a bunch of points to make a 2D area or a 3D solid. My translations are just 1 dimensional, so I haven't really done much here have I?

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in reply to @TommyTorty10's post:

joXn
@joXn

Here’s a related thought experiment that the professor who taught my probability class posed:

Suppose you have a sequence of events I1, I2, I3, … In, … that are all independent of one another (this is the kind of casual supposition mathematicians make all the time). What does that independence condition mean geometrically about the sets I1, … In …?

I can give you a hint if you need one but from the way you’re thinking I doubt you need one.

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TommyTorty10
@TommyTorty10

My first thought is that I have 2 separate diagrams. Since, the 2D diagram I've been thinking in inherently illustrates the outcomes of a singular event, then I need another diagram altogether. I could choose the place that 2nd, 2D diagram adjacent to my 1st one, just on a different plane in the 3rd dimension.

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joXn
@joXn

The thing about probabilities is that they live in an abstract space which doesn’t really have a notion of dimensionality like you’re thinking of here. It’s a measure space, which means that it’s a space whose elements are sets (not points or vectors), and it comes with a special function called the measure that can tell you the size of any of the sets; additionally because it’s a probability space we know that the measure of the entire space is equal to 1.

But that’s all we get; we don’t get “coordinates” on the sets, they don’t have “shapes” (except for implicitly, defined by their intersections), and you have to be really careful when you’re thinking about “scaling” by probabilities because if you want the result of a scaling to make sense as a probability you have to make sure that the scaling can’t give you a nonsense answer (most importantly, it can’t give you an answer that is greater than 1).

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TommyTorty10
@TommyTorty10

P(A|B) = (A∩B)/P(B) Or in other words, I'm taking this area called A and using the above formula to find a smaller area called A given B. I picture it as scaling down A by a factor that is calculated from B (and also A). Really, this just serves as a starting point for my ramble, and I'll be the first to admit that my midnight stats rambling may not be exactly coherent.

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