Cumulant-free closed-form formulas for some common (dis)similarities between densities of an exponential family

• URL: http://arxiv.org/abs/2003.02469v3
• Date: Tue, 7 Apr 2020 04:11:00 GMT
• Title: Cumulant-free closed-form formulas for some common (dis)similarities between densities of an exponential family
• Authors: Frank Nielsen and Richard Nock
• Abstract summary: In this work, we report (dis)similarity formulas which bypass the explicit use of the cumulant function. Our method requires only to partially factorize the densities canonically of the considered exponential family.
• Score: 38.13659821903422
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: It is well-known that the Bhattacharyya, Hellinger, Kullback-Leibler, $\alpha$-divergences, and Jeffreys' divergences between densities belonging to a same exponential family have generic closed-form formulas relying on the strictly convex and real-analytic cumulant function characterizing the exponential family. In this work, we report (dis)similarity formulas which bypass the explicit use of the cumulant function and highlight the role of quasi-arithmetic means and their multivariate mean operator extensions. In practice, these cumulant-free formulas are handy when implementing these (dis)similarities using legacy Application Programming Interfaces (APIs) since our method requires only to partially factorize the densities canonically of the considered exponential family.

Related papers

• Nonparametric Partial Disentanglement via Mechanism Sparsity: Sparse Actions, Interventions and Sparse Temporal Dependencies [58.179981892921056]
  This work introduces a novel principle for disentanglement we call mechanism sparsity regularization. We propose a representation learning method that induces disentanglement by simultaneously learning the latent factors. We show that the latent factors can be recovered by regularizing the learned causal graph to be sparse.
  arXiv  Detail & Related papers  (2024-01-10T02:38:21Z)
• Divergences induced by dual subtractive and divisive normalizations of exponential families and their convex deformations [7.070726553564701]
  We show that skewed Bhattacharryya distances between probability densities of an exponential family amounts to skewed Jensen divergences induced by the cumulant function. We then show how comparative convexity with respect to a pair of quasi-arithmetic means allows to deform both convex functions and their arguments.
  arXiv  Detail & Related papers  (2023-12-20T08:59:05Z)
• Causal Modeling with Stationary Diffusions [89.94899196106223]
  We learn differential equations whose stationary densities model a system's behavior under interventions. We show that they generalize to unseen interventions on their variables, often better than classical approaches. Our inference method is based on a new theoretical result that expresses a stationarity condition on the diffusion's generator in a reproducing kernel Hilbert space.
  arXiv  Detail & Related papers  (2023-10-26T14:01:17Z)
• Data-Driven Influence Functions for Optimization-Based Causal Inference [105.5385525290466]
  We study a constructive algorithm that approximates Gateaux derivatives for statistical functionals by finite differencing. We study the case where probability distributions are not known a priori but need to be estimated from data.
  arXiv  Detail & Related papers  (2022-08-29T16:16:22Z)
• Kernel Deformed Exponential Families for Sparse Continuous Attention [76.61129971916702]
  We show existence results for kernel exponential and deformed exponential families. Experiments show that kernel deformed exponential families can attend to multiple compact regions of the data domain.
  arXiv  Detail & Related papers  (2021-11-01T19:21:22Z)
• q-Paths: Generalizing the Geometric Annealing Path using Power Means [51.73925445218366]
  We introduce $q$-paths, a family of paths which includes the geometric and arithmetic mixtures as special cases. We show that small deviations away from the geometric path yield empirical gains for Bayesian inference.
  arXiv  Detail & Related papers  (2021-07-01T21:09:06Z)
• Flexible mean field variational inference using mixtures of non-overlapping exponential families [6.599344783327053]
  I show that using standard mean field variational inference can fail to produce sensible results for models with sparsity-inducing priors. I show that any mixture of a diffuse exponential family and a point mass at zero to model sparsity forms an exponential family.
  arXiv  Detail & Related papers  (2020-10-14T01:46:56Z)
• Optimal Bounds between $f$-Divergences and Integral Probability Metrics [8.401473551081748]
  Families of $f$-divergences and Integral Probability Metrics are widely used to quantify similarity between probability distributions. We systematically study the relationship between these two families from the perspective of convex duality. We obtain new bounds while also recovering in a unified manner well-known results, such as Hoeffding's lemma.
  arXiv  Detail & Related papers  (2020-06-10T17:39:11Z)

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.