All Questions
Tagged with probability-theory differential-geometry
29
questions
3
votes
0
answers
56
views
Adapted Stochastic Processes and Fiber Bundles
Suppose that we have an adapted stochastic process. In other words fix $T>0$ and suppose that for all $t \in [0,T]$ $X(t)$ is a collection of random variables, $F(t)$ is a $\sigma$-algebra, $F(s) \...
- 3,838
2
votes
0
answers
149
views
Isometry embedding
Problem:
Let $(M,g)$ be a compact Riemannian manifold. Then clearly $(M,d_R)$ is a metric space, where $$ d_R(x,y)=\|x-y\| \quad \forall x,y\in M $$
Now let's see the following:
1. The Kuratowski ...
- 21
1
vote
1
answer
661
views
Stereographic Projection of Uniform Distribution on Sphere
Consider a (punctured) unit sphere $S = \{ (x,y,z) \mid x^2+y^2+z^2 = 1, z\neq 1 \} $ and the plane $L = \{ (X,Y,0) \}$ (let us use the shorthand $(X,Y)$ for $(X,Y,0)$).
We can define the ...
- 16.7k
2
votes
0
answers
88
views
Transformation between two measures
If $\mu$ and $\nu$ are two measures, both absolutely continuous with respect to the Lebesgue measure on $\mathbb{R}^d$ with smooth densities $p_\mu(\mathbb{x})$ and $p_\nu(\mathbf{x})$, does it always ...
- 2,340
4
votes
1
answer
312
views
Isoperimetric inequality for non-spherical multivariate Gaussian
Disclaimer: Sorry in advance, if the question is not very reasonable. Recently (like a few days ago...), I've started studying isoperimetric inequalities, and my thoughts on the subject are rather ...
- 9,575
1
vote
0
answers
18
views
Defining probability of having a certain normal in surface patch.
I'm neither an expert in probability theory or in differential geometry. But for a paper I've been reading I'm trying to formalize some concept by myself. This is idea I'm trying to formalize.
...
- 5,317
2
votes
0
answers
49
views
Probability distribution on a complicated geometry.
Suppose you have a manifold with a very distorted, very complicated geometry defined by some metric tensor. However, let's say that this geometry is perhaps too complicated to predict exactly, and so ...
- 133
7
votes
1
answer
272
views
Information geometry: geometry of exponential families
I've read in various locations that the geometry of exponential families are flat.
Is this true? I don't understand because, I have also read that the family of gaussians with unknown mean and ...
- 419
3
votes
1
answer
167
views
Statistical Manifold with Non-trivial Topology
Let $(\Omega, E)$ be a measure space. An $n$-dimensional statistical model is then a tuple $(\Theta, \mathcal{M}, \Phi)$ where $\Theta \subseteq \mathbb{R}^n$ open, $\mathcal{M} = \{p_\theta := p(\...
- 3,380
2
votes
1
answer
908
views
Uniformly distributed points on spherical surface
Let $x=(x_1,\ldots,x_n)$ be uniformly distributed on the $(n-1)$-dimensional spherical surface $S^{n-1}(n^\frac{1}{2})$ of radius $n^\frac{1}{2}$. I'm trying to show that as $n\to\infty$, $x_1$ ...
- 1,354
3
votes
0
answers
67
views
Can the image of chains on a smooth manifold be thought of as a Borel $\sigma$-algebra?
Volume forms on smooth manifolds have a nice interpretation as measures, but what takes the place of the Borel $\sigma$-algebra? In particular, if we let $\mathcal{M}$ be a smooth manifold and $\...
29
votes
3
answers
6k
views
Applications of information geometry to the natural sciences
I am contemplating undergraduate thesis topics, and am searching for a topic that combines my favorite areas of analysis, differential geometry, graph theory, and probability, and that also has (...
- 293
2
votes
1
answer
140
views
Are concepts and properties studied in a category all preserved by morphisms?
When study a category, are we only interested in those concepts and properties preserved by the morphisms, not those which cannot be preserved?
For example, in Terry Tao's blog
We say that one ...
- 47.7k
5
votes
1
answer
2k
views
Integrating a $k$-dimensional (multivariate) Gaussian over a convex $k$-polytope
What is the integral of a $k$-dimensional (multivariate) Gaussian over a convex $k$-polytope?
Here I am specifically interested in $k\in\{2,3\}$, but insight on the general problem would also be ...
- 438