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
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 ...
- 1,322
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
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
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 ...
- 118k
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 ...
- 124k
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
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: $\...
- 47.4k
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 $\...
- 1,322
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 (...
- 3,965
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 ...
- 10.4k
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 $\...
- 33.2k
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 ...
- 4,878
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