Mixture and Exponential Arcs on Generalized Statistical Manifold

Andrade, Luiza Helena Félix de; Vieira, Francisca Leidmar Josué; Vigelis, Rui F.; Cavalcante, Charles C.

doi:10.3390/e20030147

Cited by 6 publications

(7 citation statements)

References 41 publications

(57 reference statements)

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“…Then, by Corollary 3, we have that c = c + u − ψ(u). The result follows immediately from[10].It follows from Corollary 1 that ϕ −1 c 2 • ϕ c 1 is of class C ∞ , and consequently, the set ϕ −1[14], Proposition 8). The relation given in the Definition 2 is an equivalence relation.…”

mentioning

confidence: 65%

“…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: 86%

“…for every w ∈ ϕ −1 In the next corollary we have that Musielak-Orlicz spaces are equal. The proof follows as the one provided in [14]. Proof.…”

Section: Proof Given Z ∈ Amentioning

confidence: 96%

“…Particular cases of that function are: q-deformed exponential and κ-exponential. In order to build this structure, as in [15], we divide P µ into equivalence classes using the connection provided by generalized exponential arcs as defined in [14]. Also, we define a set A ϕ c , that is the connected component of P µ and will be the generalized ϕ-family of probability distributions.…”

Section: Introductionmentioning

confidence: 99%

“…As a consequence, B δ (u) is contained in K ϕ c and therefore the set K ϕ c is open. The set K ϕ c defined in(14) is important to guarantee that ϕ(c + αu) may be in P µ . Now, we establish a relationship between the connection by an open arc and K ϕ c similar to that was proved in[14].…”

mentioning

confidence: 99%

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A Deformed Exponential Statistical Manifold

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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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confidence: 65%