3 ms·

I struggle with the notation, where functions are defined, and if things should be lifts/pullbacks/pushforwards. X is a real random variable. So it is a map fr

by foxes 5y ago

I struggle with the notation, where functions are defined, and if things should be lifts/pullbacks/pushforwards. X is a real random variable. So it is a map from some space of measurements, M , X : M -> R. Z is supposed to be a random variable in this case (why lower case?). f is a function from R->R, and you mean the pullback to Omega? Clearly all these spaces at least you can have some sensible notion of differential forms, differentials, so I imagine a space \Omega_M/R, so there are derivations, generalised stokes theorem etc. Then even though this is all some analytic kind of world, well we can get the algebraic structure right at least. If things are supposed to for example, Borel or whatever quantifier, I feel like that is just extra data and constraints. Stats overall seems kind of sloppy.

wvlia5 5y ago
f is a function from Rⁿ→R X is a multidimensional random variable, z is unidimensional, so that's why the difference in casing
whatshisface 5y ago
In the language of differential forms, a probability distribution would be a form of the highest order (dx ^ dy ^ dz in three dimensions), and f would be your typical map between manifolds. It would then receive identical treatment to a density, because in a sense that's what PDFs are - densities of samples in the limit of infinite samples. It is hardly sloppy, the author just wrote it in the language of vector calculus. Another treatment, the one in my textbooks, addresses it with the determinant of the Jacobian matrix from the perspective of change-of-coordinates in multidimensional integrals. As always there are as many ways to skin a cat as there are closed curves on the 2-sphere.
- r-zip 5y ago
  It is very sloppy when you come from say, analysis, where things are precisely defined, especially in more applied statistics. In my experience, any more applied than probability theory itself and you get omissions of very important details such as the probability space under consideration and the RVs involved in expectations, in addition to frequent inconsistencies and outright abuses of notation. These tendencies make it a harder subject for someone who is used to precise exposition to pick up.

by foxes 5y ago

wvlia5 5y ago

f is a function from Rⁿ→R X is a multidimensional random variable, z is unidimensional, so that's why the difference in casing

whatshisface 5y ago

In the language of differential forms, a probability distribution would be a form of the highest order (dx ^ dy ^ dz in three dimensions), and f would be your typical map between manifolds. It would then receive identical treatment to a density, because in a sense that's what PDFs are - densities of samples in the limit of infinite samples. It is hardly sloppy, the author just wrote it in the language of vector calculus. Another treatment, the one in my textbooks, addresses it with the determinant of the Jacobian matrix from the perspective of change-of-coordinates in multidimensional integrals. As always there are as many ways to skin a cat as there are closed curves on the 2-sphere.

r-zip 5y ago
It is very sloppy when you come from say, analysis, where things are precisely defined, especially in more applied statistics. In my experience, any more applied than probability theory itself and you get omissions of very important details such as the probability space under consideration and the RVs involved in expectations, in addition to frequent inconsistencies and outright abuses of notation. These tendencies make it a harder subject for someone who is used to precise exposition to pick up.