All Questions
Tagged with probability-theory differential-geometry
29
questions
5
votes
4
answers
439
views
Confusion on defining uniform distribution on hypersphere and its sampling problem
Fix a dimension $d$. Write $S^{d-1}$ for the surface of a hypersphere in $\mathbb{R}^d$, namely set of all $x = (x_1, \ldots, x_d) \in \mathbb{R}^d$ such that $|x|^2 = x_1^2 + \cdots + x_d^2 = 1$. I ...
- 2,497
0
votes
0
answers
28
views
Reference Hellinger distance as a geodesic distance
I consider a statistical manifold equipped with the Fisher Information Metric.
I want to show that for the exponential family (with no additional constraint), the Hellinger distance coincides with the ...
- 11
0
votes
0
answers
33
views
Book Recommendations for Integral Geometry, focusing on Crofton's Formula and Applications
I am seeking recommendations for books or resources that provide an engaging introduction to integral geometry, with a particular interest in Crofton's formula and its various proofs. I am looking for ...
- 1,736
0
votes
1
answer
44
views
Links between sufficient statistics and Chentsov's characterization of Fisher metric
I've been self-studying "Information Geometry" by Ay et al. fascinated by the connection between Geometry, Probability and even Statistics. The proofs are clear to me, nonetheless, even in ...
- 59
0
votes
0
answers
70
views
Integration by parts over $R^n$
Let $\mu$ be a probability measure over $\mathbb{R}^n$. Let $f$ and $g$ be two real-valued functions on $\mathbb{R}^n$. I would like to compute
$$\int_{\mathbb{R}^n} \nabla f(x) g(x) \, d\mu(x). $$
I ...
Quarth
1
vote
1
answer
64
views
Determining hyperbolic or euclidean geometry
Consider the Riemmanian product metric:
$$ ds^2= f(\theta)~d\theta^2 + f(r)~dr^2$$
for $\theta, r>0$ and $f(\theta)=2\sqrt{\theta}K_1(2\sqrt{\theta})$ where $K_1$ is a modified Bessel function.
I ...
- 756
9
votes
0
answers
211
views
Probability, potential theory and complex analysis
The connection between Markov processes and Potential Theory is well known, as is conformal invariance of Brownian motion which allows probabilistic proofs of statements in Complex Analysis, like ...
- 317
1
vote
1
answer
164
views
Why are tangent vectors in statistical manifolds $\boldsymbol e_i\approx \partial_i \log p(x,\boldsymbol\xi)$?
In section 5.1 of Amari's book, when discussing statistical manifolds, the author states that, given a manifold whose points are probability distributions, one can identify tangent vectors $\...
- 7,095
1
vote
1
answer
90
views
Probabilistic vs Geometric Theory of Integration
Motivating Question: Let $X$,$Y$ be independent standard uniform random variables. How does one show, rigorously, that
$$
\mathbb{E}[X \mid X+Y = 1] = \frac{1}{2}?
$$
I would be interested in hearing ...
- 1,003
2
votes
1
answer
101
views
Extending a probability measure on a manifold to the ambient space
Let $\pi$ be a probability measure defined on $(\mathcal{M}, \mathcal{B}(\mathcal{M}))$, where $\mathcal{M}$ is a smooth manifold and $\mathcal{B}(\mathcal{M})$ is the Borel sigma-algebra on it. Let $\...
- 5,247
1
vote
0
answers
35
views
$L^1$ almost every $t \in [0,1]$
I'm reading about curves and came across the statement that a curve $v: [0,1]\rightarrow S$ is "differentiable at $L^1$-a.e. point $t \in [0,1]$." $S$ is some metric space on which we're ...
- 2,039
5
votes
0
answers
414
views
The composition of optimal transport maps is no longer an optimal transport map
Let $X,Y,Z$ be metric spaces. Let $\mu,\nu,\omega$ be the probability measures on $X,Y,Z$, respectively. Moreover, assume all three measures vanish on small sets. Assume $T:X\to Y$ is the optimal ...
- 411
3
votes
1
answer
293
views
What is the intuition behind the Fisher-Rao metric?
I've seen the Fisher-Rao metric introduced via the following argument (see Sidhu and Kok 2019):
There is a natural pairing between the simplex and its dual space of classical random variables: $\...
- 7,095
2
votes
0
answers
45
views
Filtration generated by diffusion processes on manifolds
This is a question that bothered me for a long time. For more than in one occasion I thought I found a satisfactory answer just to realize after few months I was wrong. Consider a compact Riemannian ...
- 428
2
votes
0
answers
110
views
References for Probability Theory using Differential Forms?
Is there some known work or theory of probability that uses differential forms? I know the basics of probability, but I know there're notions of pushforward and pullback measures which they do sound ...
- 5,317
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