Approximation and Bounding Techniques for Fisher-Rao Distances

Grunnleggende konsepter

Focusing on approximation techniques for Fisher-Rao distances.

Sammendrag

The Fisher-Rao distance, a Riemannian geodesic distance induced by the Fisher information metric, is explored in this work. Various approximation and bounding techniques are discussed to calculate the Fisher-Rao distances efficiently. The article delves into closed-form calculations, generic upper bounds, and approximation schemes for different statistical models. It highlights the challenges in computing Fisher-Rao distances for complex statistical models like multivariate elliptical distributions. The importance of understanding the underlying geometric structure of Fisher-Rao distances is emphasized to facilitate accurate approximations.

Statistikk

The Jeffreys divergence between centered d-variate Gaussians pµ,Σ0 and pµ,Σ1 is DJ(pµ, Σ0, pµ,Σ1) = 1/2tr(Σ−1₁ Σ₀ + Σ−1₀ Σ₁) - d.
The infinitesimal Fisher length element for centered d-variate normal distributions is ds²Σ(Σ, dΣ) = 1/2tr((Σ⁻¹dΣ)²).
The Fisher information matrix is invariant under reparameterization of the parameter space.
The Riemannian geodesics can be calculated in closed-form when the Christoffel symbols vanish or when the connection is flat.
The GeomStats software package provides a generic implementation of the Fisher-Rao distance using automatic differentiation.

Sitater

"The notion of dissimilarity between two probability distributions is essential in statistics." - [38]
"In practice, a simple test on the Fisher information matrix allows to check whether the corresponding Fisher metric is a Hessian metric or not." - [49]
"Stigler noticed that Hotelling historically first considered the Fisher-Rao distance." - [54]

Viktige innsikter hentet fra

Approximation and bounding techniques for the Fisher-Rao distances

by Frank Nielse... klokken arxiv.org 03-18-2024

https://arxiv.org/pdf/2403.10089.pdf

Dypere Spørsmål

How do different approximation methods impact the accuracy of calculating Fisher-Rao distances

Different approximation methods can impact the accuracy of calculating Fisher-Rao distances in various ways.

Approximating the Fisher-Rao lengths of curves: This method involves approximating the length of a smooth curve connecting two points on the Fisher-Rao manifold. While this method may provide a quick approximation, it relies on discretization and f-divergences, which can introduce errors depending on the chosen step size and interpolation technique.

Using closed-form geodesics: When closed-form expressions for Fisher-Rao geodesics are available, they offer an exact calculation of the distance between two points. However, these calculations may be computationally intensive or not feasible for complex statistical models.

Fisher-Manhattan upper bound: This method uses a hypercube graph to approximate the shortest path between two points on the manifold. While it provides an upper bound on the distance, it may not always capture the true geodesic path accurately.
The choice of approximation method depends on factors such as computational resources, model complexity, and desired level of accuracy. Each method has its strengths and limitations in terms of speed, precision, and ease of implementation.

What are potential limitations of using f-divergences as an approximation technique for Fisher-Rao lengths

Using f-divergences as an approximation technique for Fisher-Rao lengths has some potential limitations:

Sensitivity to parameter choices: The accuracy of approximations using f-divergences can depend heavily on selecting appropriate parameters such as step sizes or interpolation functions. Poor choices may lead to significant errors in estimating distances.

Validity only for certain distributions: F-divergences are suitable for specific types of probability distributions where they have known properties (e.g., convexity). For more complex or non-standard distributions, f-divergences may not provide accurate approximations.

Assumptions about smoothness: F-divergence-based approximations assume that probability densities are smooth and well-behaved. In cases where distributions exhibit irregularities or discontinuities, these methods may yield inaccurate results.

How can insights from group actions and maximal invariants enhance our understanding of statistical transformation models

Insights from group actions and maximal invariants can enhance our understanding of statistical transformation models by providing structural insights into their properties:

Group action perspective: By considering how groups act on statistical models through transformations like reparameterizations or symmetries, we gain a deeper understanding of how different parameterizations relate to each other geometrically. This perspective helps identify invariant structures within models under group actions.

Maximal invariant concept: Maximal invariants highlight key features that remain unchanged under certain transformations within a statistical model. Understanding these maximal invariants allows us to identify essential characteristics that define equivalence classes within transformation models.
By leveraging insights from group theory and maximal invariants, we can uncover hidden symmetries and relationships within statistical transformation models that go beyond traditional approaches based solely on parameter spaces or likelihood functions.