Approximation and bounding techniques for the Fisher-Rao distances between parametric statistical models

• URL: http://arxiv.org/abs/2403.10089v3
• Date: Wed, 22 May 2024 00:45:47 GMT
• Title: Approximation and bounding techniques for the Fisher-Rao distances between parametric statistical models
• Authors: Frank Nielsen,
• Abstract summary: We consider several numerically robust approximation and bounding techniques for the Fisher-Rao distances. In particular, we obtain a generic method to guarantee an arbitrarily small additive error on the approximation. We propose two new distances based either on the Fisher-Rao lengths of curves serving as proxies of Fisher-Rao geodesics.
• Score: 7.070726553564701
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The Fisher-Rao distance between two probability distributions of a statistical model is defined as the Riemannian geodesic distance induced by the Fisher information metric. In order to calculate the Fisher-Rao distance in closed-form, we need (1) to elicit a formula for the Fisher-Rao geodesics, and (2) to integrate the Fisher length element along those geodesics. We consider several numerically robust approximation and bounding techniques for the Fisher-Rao distances: First, we report generic upper bounds on Fisher-Rao distances based on closed-form 1D Fisher-Rao distances of submodels. Second, we describe several generic approximation schemes depending on whether the Fisher-Rao geodesics or pregeodesics are available in closed-form or not. In particular, we obtain a generic method to guarantee an arbitrarily small additive error on the approximation provided that Fisher-Rao pregeodesics and tight lower and upper bounds are available. Third, we consider the case of Fisher metrics being Hessian metrics, and report generic tight upper bounds on the Fisher-Rao distances using techniques of information geometry. Uniparametric and biparametric statistical models always have Fisher Hessian metrics, and in general a simple test allows to check whether the Fisher information matrix yields a Hessian metric or not. Fourth, we consider elliptical distribution families and show how to apply the above techniques to these models. We also propose two new distances based either on the Fisher-Rao lengths of curves serving as proxies of Fisher-Rao geodesics, or based on the Birkhoff/Hilbert projective cone distance. Last, we consider an alternative group-theoretic approach for statistical transformation models based on the notion of maximal invariant which yields insights on the structures of the Fisher-Rao distance formula which may be used fruitfully in applications.

Related papers

• Relative-Translation Invariant Wasserstein Distance [82.6068808353647]
  We introduce a new family of distances, relative-translation invariant Wasserstein distances ($RW_p$) We show that $RW_p distances are also real distance metrics defined on the quotient set $mathcalP_p(mathbbRn)/sim$ invariant to distribution translations.
  arXiv  Detail & Related papers  (2024-09-04T03:41:44Z)
• von Mises Quasi-Processes for Bayesian Circular Regression [57.88921637944379]
  We explore a family of expressive and interpretable distributions over circle-valued random functions. The resulting probability model has connections with continuous spin models in statistical physics. For posterior inference, we introduce a new Stratonovich-like augmentation that lends itself to fast Markov Chain Monte Carlo sampling.
  arXiv  Detail & Related papers  (2024-06-19T01:57:21Z)
• The Fisher-Rao geometry of CES distributions [50.50897590847961]
  The Fisher-Rao information geometry allows for leveraging tools from differential geometry. We will present some practical uses of these geometric tools in the framework of elliptical distributions.
  arXiv  Detail & Related papers  (2023-10-02T09:23:32Z)
• Fisher-Rao distance and pullback SPD cone distances between multivariate normal distributions [7.070726553564701]
  We introduce a class of distances based on diffeomorphic embeddings of the normal manifold into a submanifold. We show that the projective Hilbert distance on the cone yields a metric on the embedded normal submanifold. We show how to use those distances in clustering tasks.
  arXiv  Detail & Related papers  (2023-07-20T07:14:58Z)
• A numerical approximation method for the Fisher-Rao distance between multivariate normal distributions [12.729120803225065]
  We use discretizing curves joining normal distributions and approximating Rao's distances between successive nearby normal distributions on the curves by the square root of Jeffreys divergence. We report on our experiments and assess the quality of our approximation technique by comparing the numerical approximations with both lower and upper bounds.
  arXiv  Detail & Related papers  (2023-02-16T09:44:55Z)
• Limits on Parameter Estimation of Quantum Channels [0.0]
  We study the task of estimating unknown parameters encoded in a quantum channel in the sequential setting. Our goal is to establish lower bounds (called Cramer-Rao bounds) on the estimation error.
  arXiv  Detail & Related papers  (2022-01-05T17:59:04Z)
• Near-optimal estimation of smooth transport maps with kernel sums-of-squares [81.02564078640275]
  Under smoothness conditions, the squared Wasserstein distance between two distributions could be efficiently computed with appealing statistical error upper bounds. The object of interest for applications such as generative modeling is the underlying optimal transport map. We propose the first tractable algorithm for which the statistical $L2$ error on the maps nearly matches the existing minimax lower-bounds for smooth map estimation.
  arXiv  Detail & Related papers  (2021-12-03T13:45:36Z)
• Physics-Informed Machine Learning Method for Large-Scale Data Assimilation Problems [48.7576911714538]
  We extend the physics-informed conditional Karhunen-Lo'eve expansion (PICKLE) method for modeling subsurface flow with unknown flux (Neumann) and varying head (Dirichlet) boundary conditions. We demonstrate that the PICKLE method is comparable in accuracy with the standard maximum a posteriori (MAP) method, but is significantly faster than MAP for large-scale problems.
  arXiv  Detail & Related papers  (2021-07-30T18:43:14Z)
• On Projection Robust Optimal Transport: Sample Complexity and Model Misspecification [101.0377583883137]
  Projection robust (PR) OT seeks to maximize the OT cost between two measures by choosing a $k$-dimensional subspace onto which they can be projected. Our first contribution is to establish several fundamental statistical properties of PR Wasserstein distances. Next, we propose the integral PR Wasserstein (IPRW) distance as an alternative to the PRW distance, by averaging rather than optimizing on subspaces.
  arXiv  Detail & Related papers  (2020-06-22T14:35:33Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.