Questions tagged [information-geometry]
Information geometry is a branch of mathematics that applies the techniques of differential geometry to the field of probability theory. This is done by taking probability distributions for a statistical model as the points of a Riemannian manifold, forming a statistical manifold. The Fisher information metric provides the Riemannian metric.
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Research situation in the field of Information Geometry
I am now doing an article survey on the field of information geometry started by S.Amari and Barndorff-Nielson. I want to know some research situation in this field.
I have read (4) and parts of (3). ...
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Relation between information geometry and geometric deep learning
Disclaimer: This is a cross-post from a very similar question on math.SE. I allowed myself to post it here after reading this meta
post about cross-posting between mathoverflow and math.SE, I did
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Which books should I read in order to be prepared to study information geometry?
At the moment, I am preparing my master's thesis (in statistics) and I intend to keep studying in order to pursue a doctoral degree. To be precise, I am mainly interested in studying Information ...
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Reviews of Probability in High Dimension not by Van Handel
I'm completely in love with Ramon van Handel's lecture notes Probability in High Dimension and I would like to find more learning resources. Lecture notes or reviews would be ideal as anything in this ...
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Upper-bound on the Fisher-Rao distance between multivariate Gaussian measures by the KL-divergence
Let $\mu$ and $\nu$ be two multivariate Gaussian measures on $\mathbb{R}^d$ with non-singular covariance matrices. Can the Fisher-Rao distance $d(\mu,\nu)$ computed on the information manifold of non-...
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Approximation of Wasserstein distance between $p_\theta$ and $p_{\theta + d\theta}$
Given a parametric family of distributions $\{p_\theta\mid\theta \in \Theta\}$, one can show that under some regularity conditions, the following approximation is valid
$$\operatorname{KL}(p_\theta\...
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Introduction to information geometry and/or geometric control theory
Some background: I'vebeen searching for a research project to work through my grad studies and I found information geometry like a strong candidate but the amount of work out there is overwhelming. I ...
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What is the correct notion of morphism between statistical manifolds?
Given two statistical manifolds, is there a notion of "isomorphic"? What are morphisms?
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Singular Fisher information matrix and existence of unbiased estimators
I'm doing some research into the Cramer-Rao bound for time of arrival localization and have come across a rather strange result: the FIM is singular, but there exists an unbiased estimator. My ...
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Projections in infinite dimensional statistical manifolds
I'm struggling to understand the geometry of projection for infinite dimensional statistical manifolds. In finite dimensions, a strictly convex smooth function $F$ defines a Bregman divergence. From ...
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reverse KL-divergence: Bregman or not?
I am having a little trouble getting my head around the two "directions" of the Kullback-Leibler divergence:
Definition (Kullback-Leibler divergence) For discrete probability distributions $...
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Is the gradient of a strictly convex, continuously differentiable function a homeomorphism?
Let $X\subseteq\mathbb{R}^n$ be a convex set. Let $f:X\to\mathbb{R}$ be a strictly convex function that is differentiable on the (non-empty) relative interior of $X$.
$\nabla f$ is a bijection, but is ...
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Smallest $\mathrm{D}(Q\|P)$ given fixed marginals $\mathrm{D}(Q_X\|P_X)$ and $\mathrm{D}(Q_Y\|P_Y)$
Let $P$ be a distribution on a set $U\times V$ with marginal distributions $P_X$ and $P_Y$.
Suppose we have two values $d_x, d_y\in\mathbb R$, and we want to find the distribution $Q$ absolutely ...
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Covariance operator analogue for manifolds and respective measure manifolds
Assume $E$ is a connected riemannian manifold with geodesic metric space structure given by $d$ and $P$ is a probability measure over $E$ with Borel sigma-algebra given by this metric structure. Also ...
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Comparison of Information and Wasserstein Topologies
There are many possible metrics one can place on the space of Gaussian probability measures on $\mathbb{R}^n$, with strictly positive definite co-variance matrices. Let's denote this space by $X$.
I'...
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How to study to learn differential geometry for applying it to statistics
Basically I want to learn information geometry or specifically the application of differential geometry in statistics to do a project. I am from a statistical background and have a knowledge about ...
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Relationship between $\alpha$-divergences?
I am working with $\alpha$-divergences and was wondering how understand the relationship between the definitions of Renyi and Amari?
Renyi:
$D_{\alpha}[p||q] = \frac{1}{\alpha - 1} \log \int p^{\...
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Pythagorean theorems for other distances
Question
The usual projection in $\mathbb{R}^n$ on a subspace can be defined as the point that minimizes the squared distance to the subspace. I'll call the Pythagorean theorem the easy fact that, ...
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Divergence for Bhattacharya Information matrix
The Fisher information matrix (in the scalar parameter case) can be obtained from the Kullback-Leibler divergence by
$$g(\theta) = -\frac{\partial}{\partial \theta}\frac{\partial}{\partial \theta'}D(...
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Distance measures that preserve Pythagoras' theorem but break the triangle inequality
In information geometry, we can think of the Kullback-Leibler divergence as being "something like a squared distance."
The sense of this is that if we have three probability measures, $P$, $Q$ and $R$...
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Gromov hyperbolicity for (non-geodesic) metrics on the upper-half plane invariant with respect to SL(2, R) action
$\DeclareMathOperator\SL{SL}$Let $d$ be a metric on the upper-half plane $\mathbb H = \{(x,y) : y > 0\}$ which is invariant with respect to the action of $\SL(2, \mathbb R)$ to $\mathbb H$ which is ...
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About optimization with Renyi divergence
Can someone link me to some pedagogic example of computing the Renyi divergence between two discrete/continuous distributions? Like examples where someone has been able to obtain a neat closed form or ...
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Information monotonicity of divergence => function of $f$-divergence
It is well-known that $f$-divergences defined on $\mathcal P(\mathcal X)$ where $\mathcal X$ is a measure space with $\sigma$-algebra $\mathcal B$ satisfy the property of information monotonicity:
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Exponential families closed under affine transformations
Let $(\Omega,\Sigma,\mu)$ be a probability space and let $\mathcal{M}$ be an exponential family of probability distributions for $\mu$ of the following form: There are $\varphi_1,\dots,\varphi_n:\...
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Direct calculation of the Fisher distance via Riemannian geodesics
I'm looking for a reference for a direct calculation of the Fisher distance (to avoid overloading the term "metric") $d_F(x,y) := 2 \cos^{-1} \sum_i \sqrt{x_i y_i}$ as the geodesic distance ...
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Conditions for: (local) lipschitz stability of I-projection
The following post builds on this post; I'll begin by quoting the setting.
Background from Previous Question:
$\newcommand\SS{P}\newcommand\TT{Q}$Call a Gaussian probability measure $\SS$ on $\mathbb{...
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Interpolation inequality related to the 5/3-Laplace operator
I'm having trouble with an estimate that would be helpful in information geometry.
The background is the following. Suppose we have a smooth positive function $g:X \to \mathbb{R}^+$ where $X$ is a ...
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From $f$-divergence to its dual: the transformation of convex functions on $\mathbb R_+$ by $f^*(t) = 1 f(\frac 1 t)$
I would like to understand the relationship between minimising the KL divergence $P \mapsto D_{KL}[P,Q]$ and the reverse KL divergence $P\mapsto D_{KL}^*[P,Q]=D_{KL}[Q,P]$ for probability measures $P$ ...
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Parametric distances on product spaces of measures
Disclaimer: Please excuse my loose language. I'm neither an expert in geometry nor probability. Please ask for clarification if something appears unclear or awkward to you.
Let $X$ be a topological ...
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Partial information decomposition for tangle machines
In (Williams and Beer, 2010), they define the partial information decomposition (PID) as a generalization of Shannon's Mutual Information for multiple information sources. Their key insight is that ...
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Is Bregman divergence independent of coordinates?
Question
Is Bregman divergence free of coordinates?
Although it is invariant w.r.t. which local affine coordinate you take, is it possible to prove that it does not change w.r.t. an arbitrary change ...
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Complete statistical manifolds
Here, by a statistical manifold I mean a $d$-dimensional Riemannian manifold whose points are probability measures on $\mathbb{R}^n$. What are some well-studied/interesting examples of statistical ...
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Ideas on how to prove Pythagorean identity involving Wasserstein distances?
I conjectured earlier that if $P$ and $Q$ were two probability measures, then we could show
$$W^2(P,Q) = \min_{T} [d^2(P,T_{\#}P) + W^2(T_{\#}P,Q)]$$ where $W^2(P,Q)$ denotes the squared Wasserstein-2 ...
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How much can KL divergence decrease by diluting the reference distribution
Let $\Omega$ be a countable set and $\mu,\nu\colon\Omega\to[0,1]$ be distributions on $\Omega$, that is we have $\sum_{x\in\Omega}\mu(x)=1$ and likewise for $\nu$. The Kullback-Leibler divergence of $\...
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What's meant by this measure in the sample space of a probability distribution?
In Amari's "Information Geometry and its Applications", in section 2.1 they define the exponential family as
$$ p(x, \pmb\theta) = \exp \{ \theta^i h_i (x) + k(x) - \psi (\pmb\theta) \}$$
(...
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Convexity of exponential family
It is known that (given a $\sigma$-finite Borel reference measure $\nu$ on $\mathbb{R}$) the parameter space of an exponential family is convex in Euclidean space. However, my question is, for an the ...
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(quasi)metric on Riemannian manifolds via Brownian Motion?
Given points $a$ and $b$ on a Riemannian manifold $M$, I would like a (quasi)metric that corresponds to some property of Brownian Motion from $a$ to $b$ (or rather, to $\epsilon$-ball $B = \{ x : |x - ...
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Joint lower semicontinuity of the Rényi divergence in all three arguments
Let $X$ be a standard Borel space (I'm already interested in the case where $X$ is finite, i.e., $X=\lbrace 1,\cdots,n\rbrace$). Let $P,Q$ be probability measures on $X$ such that $P\ll Q$. Then the ...
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Is there any correspondence between Jacobi forms and automorphic forms on the unit ball in $\mathbb{C}^2$?
Apologies in advance if this question is obvious (or obviously false); number theory is far from my area of expertise. Let me state my questions and then I'll explain the motivation for asking them.
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Rotationally symmetric manifold with statistical structure
Statistical manifolds are riemannian manifolds endowed with an affine torsion-free connection which is not necessarily Levi-Civita but verifies a weaker compatibility condition.
More precisely, a ...
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Inequality on the Kullback-Leibler divergence
Let us define the arithmetic, geometric, and harmonic means of $x,y \in \mathbb{R}$ weighted by $\alpha =(\alpha_x,\alpha_y) \in [0,1]$, respectively as
\begin{equation}
a_\alpha(x,y) = \frac{\...
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Connecting Wasserstein distance with mutual information?
Suppose I have Markov chains:
$$X \rightarrow f(X) \rightarrow g(X)$$
$$Y \rightarrow f(Y) \rightarrow g(Y)$$
where it is known that minimizing the $\mathbb{E}(g(X)) - \mathbb{E}(g(Y))$ minimizes the ...
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Does a 1-Lipschitz function preserve mutual information between two random variables?
Suppose we have a 1-Lipschitz function $f$ such that 1-Lipschitzness is preserved, with $D_A(f(X), f(Y)) \leq D_B(X, Y)$ for some metric spaces $A$ and $B$.
Does this also imply that $I(f(X); f(Y)) = ...
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Is there any geometric interpretation for the trace of Fisher information matrix?
Consider a parametric family $p_\theta(x)$ of distributions, with parameter $\theta \in \Theta \subseteq \mathbb R^p$.
If the mapping $\theta \mapsto p_\theta(x)$ is continuously differentiable at $\...
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Variational inference: Does the natural gradient follow (Fisher-Rao) geodesics locally?
Amari's natural gradient descent is a well-known optimisation algorithm for functionals defined on statistical manifolds. It consists of preconditioning the vanilla gradient descent update rule with ...
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Minimizer of a class of SDEs
Setup
Let $\mathscr{H}$ be a separable Hilbert space, $\mathcal{X}\triangleq \langle \Omega,\mathscr{F},\mathscr{F}_t,\mathbb{P}\rangle$ be a stochastic base and $X_t$ be an $H$-valued stochastic ...
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References: error and stability estimates for information projection
$\newcommand\SS{P}\newcommand\TT{Q}$I will call a Gaussian probability measure $\SS$ on $\mathbb{R}^d$ isotropic if its covariance matrix is diagonal with non-vanishing determinant; i.e. $\Sigma_{i,i}&...
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Relationship between the Fisher distance and Kulback Leibler divergence
I am reading the 2017 book "Information geometry" by Ay, Jost, Lê, Schwachhöfer. The Fisher distance is given by
$$
d^F(\mu, \nu) := \inf_{\gamma} L(\gamma)
$$
for curves $\gamma:[0,1]\to P$ ...
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Parameterization of exponential family
Let $\{\mathbb{P}_{\theta}\}_{\theta}$ be an exponential family of probability measures, all with finite mean. Under what conditions is the parameterization map $\theta\mapsto \mathbb{P}_{\theta}$ ...
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Representation theory for symmetries of probability distribution functions
I would like to parameterize all the possible modifications to a probability density function. Is there a representation theory for this? Something along the lines of, these are all the operators $L$ ...