On the Existence of Paths Connecting Probability Distributions

Vigelis, Rui F.; Andrade, Luiza Helena Félix de; Cavalcante, Charles C.

doi:10.1007/978-3-319-68445-1_92

Cited by 4 publications

(3 citation statements)

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“…, (27) where κ(α) satisfies the Equation (12). This generalization in the case α ∈ {0, 1} is defined as the limit R,ϕ (p z) = D ϕ (p z) < ∞.…”

Section: Generalization Of Divergence Of Rényi and Exp κmentioning

confidence: 99%

“…Furthermore, the arcs for the deformed exponential function were investigated and it was provided the necessary and sufficient conditions to connect by a ϕ-arc any two probability distributions [11]. This result was generalized later by [12,13]. A generalization to exponential arcs was defined in [14] and it also proved that the probability distribution z belongs to the ϕ-family F ϕ c if, and only if, z is connected to p by an open ϕ-arc.…”

Section: Introductionmentioning

confidence: 99%

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A Deformed Exponential Statistical Manifold

Vieira

Andrade

Vigelis

et al. 2019

Entropy

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Consider μ a probability measure and P μ the set of μ -equivalent strictly positive probability densities. To endow P μ with a structure of a C ∞ -Banach manifold we use the φ -connection by an open arc, where φ is a deformed exponential function which assumes zero until a certain point and from then on is strictly increasing. This deformed exponential function has as particular cases the q-deformed exponential and κ -exponential functions. Moreover, we find the tangent space of P μ at a point p, and as a consequence the tangent bundle of P μ . We define a divergence using the q-exponential function and we prove that this divergence is related to the q-divergence already known from the literature. We also show that q-exponential and κ -exponential functions can be used to generalize of Rényi divergence.

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“…, (27) where κ(α) satisfies the Equation (12). This generalization in the case α ∈ {0, 1} is defined as the limit R,ϕ (p z) = D ϕ (p z) < ∞.…”

Section: Generalization Of Divergence Of Rényi and Exp κmentioning

confidence: 99%