All Questions
Tagged with probability-theory differential-geometry
29
questions
4
votes
4
answers
420
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 ...
- 2,478
0
votes
0
answers
27
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 ...
- 1,698
0
votes
1
answer
43
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 ...
- 946
9
votes
0
answers
207
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
150
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 ...
- 983
2
votes
1
answer
100
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 ...
- 2,029
5
votes
0
answers
409
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
286
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