Decision Theory and large deviations for dynamical hypotheses tests:
The Neyman-Pearson Lemma, Min-Max and Bayesian tests

Ferreira, Hermes H.; Lopes, Artur O.; Lopes, Sílvia da Silva

doi:10.3934/jdg.2021031

Cited by 5 publications

(11 citation statements)

References 25 publications

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“…The probability µ λ corresponds in [40] to the concept of mixture distribution. In [23] Bayesian Hypothesis Tests are considered for the family described by (15).…”

Section: Kl-divergence and Dynamical Information Projectionsmentioning

confidence: 99%

“…From a Bayesian point of view, the probability µ 1 describes the prior probability and µ λ plays the role of the posterior probability in the inductive inference problem described by expression D KL (µ λ , µ 1 ) (see Section 2.10 in [10], [37] and [20]). The function log J λ − log J 1 should be considered as the likelihood function (see [23]).…”

Section: Kl-divergence and Dynamical Information Projectionsmentioning

confidence: 99%

“…A natural question to ask is when these inequalities appear when considering some extremality property regarding a fixed convex set of probabilities (like expression (23) in the First problem to be defined next).…”

Section: Kl-divergence and Dynamical Information Projectionsmentioning

confidence: 99%

“…and we denote by µ λ the Gibbs probability associated to the Jacobian J λ . When J 0 ∈ Θ 1 minimize the Kullback-Leibler divergence in problem (23), and J λ satisfies (25), we get for any…”

Section: Kl-divergence and Dynamical Information Projectionsmentioning

confidence: 99%

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Geodesics and dynamical information projections on the manifold of Hölder equilibrium probabilities

Lopes¹,

Ruggiero²

2022

Preprint

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We consider here the discrete time dynamics described by a transformation T : M → M , where T is either the action of shift T = σ on the symbolic space M = {1, 2, ..., d} N , or, T describes the action of a d to 1 expanding transformation T : S 1 → S 1 of class C 1+α ( for example x → T (x) = d x (mod 1) ), where M = S 1 is the unit circle. It is known that the infinite-dimensional manifold N of equilibrium probabilities for Hölder potentials A : M → R is an analytical manifold and carries a natural Riemannian metric associated with the asymptotic variance. We show here that under the assumption of the existence of a Fourier-like Hilbert basis for the kernel of the Ruelle operator there exists geodesics paths. When T = σ and M = {0, 1} N such basis exists.In a different direction, we also consider the KL-divergence D KL (µ 1 , µ 2 ) for a pair of equilibrium probabilities. If D KL (µ 1 , µ 2 ) = 0, then µ 1 = µ 2 . Although D KL is not a metric in N , it describes the proximity between µ 1 and µ 2 . A natural problem is: for a fixed probability µ 1 ∈ N consider the probability µ 2 in a convex set of probabilities in N which minimizes D KL (µ 1 , µ 2 ). This minimization problem is a dynamical version of the main issues considered in information projections. We consider this problem in N , a case where all probabilities are dynamically invariant, getting explicit equations for the solution sought. Triangle and Pythagorean inequalities will be investigated.

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“…The probability µ λ corresponds in [40] to the concept of mixture distribution. In [23] Bayesian Hypothesis Tests are considered for the family described by (15).…”

Section: Kl-divergence and Dynamical Information Projectionsmentioning

confidence: 99%

Section: Kl-divergence and Dynamical Information Projectionsmentioning

confidence: 99%

Section: Kl-divergence and Dynamical Information Projectionsmentioning

confidence: 99%

Section: Kl-divergence and Dynamical Information Projectionsmentioning

confidence: 99%

See 2 more Smart Citations

Geodesics and dynamical information projections on the manifold of Hölder equilibrium probabilities

Lopes¹,

Ruggiero²

2022

Preprint

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“…Large deviations estimates are commonly used in decision theory (see e.g. [5,18,37]) In the context of dynamical systems, the exponential rate of convergence in large deviations are defined in terms of rate functions, often described by thermodynamic quantities as pressure and entropy. In the case of level-1 large deviation estimates these can be defined as follows.…”

Section: 3mentioning

confidence: 99%

Bayes posterior convergence for loss functions via almost additive Thermodynamic Formalism

Lopes¹,

Lopes²,

Varandas³

2020

Preprint

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Statistical inference can be seen as information processing involving input information and output information that updates belief about some unknown parameters. We consider the Bayesian framework for making inferences about dynamical systems from ergodic observations, where the Bayesian procedure is based on the Gibbs posterior inference, a decision process generalization of standard Bayesian inference (see [3,22]) where the likelihood is replaced by the exponential of a loss function. In the case of direct observation and almost-additive loss functions, we prove an exponential convergence of the a posteriori measures a limit measure. Our estimates on the Bayes posterior convergence for direct observation are related but complementary to those in [30]. Our approach makes use of non-additive thermodynamic formalism and large deviation properties [24,25,38] instead of joinings.

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Bayes Posterior Convergence for Loss Functions via Almost Additive Thermodynamic Formalism

Lopes

Varandas

2022

J Stat Phys

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Statistical inference can be seen as information processing involving input information and output information that updates belief about some unknown parameters. We consider the Bayesian framework for making inferences about dynamical systems from ergodic observations, where the Bayesian procedure is based on the Gibbs posterior inference, a decision process generalization of standard Bayesian inference (see [ 7 , 37 ]) where the likelihood is replaced by the exponential of a loss function. In the case of direct observation and almost-additive loss functions, we prove an exponential convergence of the a posteriori measures to a limit measure. Our estimates on the Bayes posterior convergence for direct observation are related and extend those in [ 47 ] to a context where loss functions are almost-additive. Our approach makes use of non-additive thermodynamic formalism and large deviation properties [ 39 , 40 , 57 ] instead of joinings.

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Decision Theory and large deviations for dynamical hypotheses tests: The Neyman-Pearson Lemma, Min-Max and Bayesian tests

Cited by 5 publications

References 25 publications

Geodesics and dynamical information projections on the manifold of Hölder equilibrium probabilities

Geodesics and dynamical information projections on the manifold of Hölder equilibrium probabilities

Bayes posterior convergence for loss functions via almost additive Thermodynamic Formalism

Bayes Posterior Convergence for Loss Functions via Almost Additive Thermodynamic Formalism

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