Normalization and ϕ-function: definition and consequences

07/11/2017
Publication GSI2017
OAI : oai:www.see.asso.fr:17410:22626
contenu protégé  Document accessible sous conditions - vous devez vous connecter ou vous enregistrer pour accéder à ou acquérir ce document.
- Accès libre pour les ayants-droit
 

Résumé

It is known from the literature that a ϕ-function may be used to construct the ϕ -families of probability distributions. In this paper, we assume that one of the properties in the definition of ϕ -function is not satisfied and we analyze the behavior of the normalizing function near the boundary of its domain. As a consequence, we find a measurable function that does not belong to the Musielak–Orlicz class, but the normalizing function applied to this found function converges to a finite value near the boundary of its domain. We conclude showing that this change in the definition of ϕ -function affects the behavior of the normalizing function.

Normalization and ϕ-function: definition and consequences

Collection

application/pdf Normalization and ϕ-function: definition and consequences (slides)
application/pdf Normalization and ϕ-function: definition and consequences Luiza Helena Félix de Andrade, Rui F. Vigelis, Francisca L. J. Vieira, Charles Casimiro Cavalcante
Détails de l'article
contenu protégé  Document accessible sous conditions - vous devez vous connecter ou vous enregistrer pour accéder à ou acquérir ce document.
- Accès libre pour les ayants-droit

It is known from the literature that a ϕ-function may be used to construct the ϕ -families of probability distributions. In this paper, we assume that one of the properties in the definition of ϕ -function is not satisfied and we analyze the behavior of the normalizing function near the boundary of its domain. As a consequence, we find a measurable function that does not belong to the Musielak–Orlicz class, but the normalizing function applied to this found function converges to a finite value near the boundary of its domain. We conclude showing that this change in the definition of ϕ -function affects the behavior of the normalizing function.
Normalization and ϕ-function: definition and consequences

Média

Voir la vidéo

Métriques

0
0
133.93 Ko
 application/pdf
bitcache://b1e3d2ff2c0bdbd693f0a9c76774e708d5ecec74

Licence

Creative Commons Aucune (Tous droits réservés)

Sponsors

Sponsors Platine

alanturinginstitutelogo.png
logothales.jpg

Sponsors Bronze

logo_enac-bleuok.jpg
imag150x185_couleur_rvb.jpg

Sponsors scientifique

logo_smf_cmjn.gif

Sponsors

smai.png
gdrmia_logo.png
gdr_geosto_logo.png
gdr-isis.png
logo-minesparistech.jpg
logo_x.jpeg
springer-logo.png
logo-psl.png

Organisateurs

logo_see.gif
<resource  xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"
                xmlns="http://datacite.org/schema/kernel-4"
                xsi:schemaLocation="http://datacite.org/schema/kernel-4 http://schema.datacite.org/meta/kernel-4/metadata.xsd">
        <identifier identifierType="DOI">10.23723/17410/22626</identifier><creators><creator><creatorName>Rui F. Vigelis</creatorName></creator><creator><creatorName>Charles Casimiro Cavalcante</creatorName></creator><creator><creatorName>Luiza Helena Félix de Andrade</creatorName></creator><creator><creatorName>Francisca L. J. Vieira</creatorName></creator></creators><titles>
            <title>Normalization and ϕ-function: definition and consequences</title></titles>
        <publisher>SEE</publisher>
        <publicationYear>2018</publicationYear>
        <resourceType resourceTypeGeneral="Text">Text</resourceType><dates>
	    <date dateType="Created">Fri 9 Mar 2018</date>
	    <date dateType="Updated">Fri 9 Mar 2018</date>
            <date dateType="Submitted">Mon 17 Dec 2018</date>
	</dates>
        <alternateIdentifiers>
	    <alternateIdentifier alternateIdentifierType="bitstream">b1e3d2ff2c0bdbd693f0a9c76774e708d5ecec74</alternateIdentifier>
	</alternateIdentifiers>
        <formats>
	    <format>application/pdf</format>
	</formats>
	<version>37378</version>
        <descriptions>
            <description descriptionType="Abstract">
It is known from the literature that a ϕ-function may be used to construct the ϕ -families of probability distributions. In this paper, we assume that one of the properties in the definition of ϕ -function is not satisfied and we analyze the behavior of the normalizing function near the boundary of its domain. As a consequence, we find a measurable function that does not belong to the Musielak–Orlicz class, but the normalizing function applied to this found function converges to a finite value near the boundary of its domain. We conclude showing that this change in the definition of ϕ -function affects the behavior of the normalizing function.

</description>
        </descriptions>
    </resource>
.

Normalization and ϕ-function: definition and consequences Luiza H. F. de Andrade1 , Rui F. Vigelis2 , Francisca L. J. Vieira3 , and Charles C. Cavalcante4 1 Center of Exact and Natural Sciences, Federal Rural University of Semi-Arid Region, Mossoró-RN, Brazil, luizafelix@ufersa.edu.br. 2 Computer Engineering, Campus Sobral, Federal University of Ceará, Sobral-CE, Brazil, rfvigelis@ufc.br. 3 Department of Mathematics, Regional University of Cariri, Juazeiro do Norte-CE, leidmar.vieira@urca.br. 4 Department of Teleinformatics Engineering, Federal University of Ceará, Fortaleza-CE, Brazil, charles@ufc.br. Abstract. It is known from the literature that a ϕ-function may be used to construct the ϕ-families of probability distributions. In this paper, we assume that one of the properties in the definition of ϕ-function is not satisfied and we analyze the behavior of the normalizing function near the boundary of its domain. As a consequence, we find a measurable function that does not belong to the Musielak–Orlicz class, but the normalizing function applied to this found function converges to a finite value near the boundary of its domain. We conclude showing that this change in the definition of ϕ-function affects the behavior of the normalizing function. 1 Introduction In [7] was obtained a generalization of exponential families of probability dis- tributions [5,4], called ϕ-families. The construction of these families is based on Musielak–Orlicz spaces [2] and on a function, called ϕ-function, which satisfies some properties. Another generalization of exponential families of probability distributions in infinite-dimensional setting was studied in [8]. In [6], it was studied the ∆2-condition and its consequences on ϕ-families of probability dis- tributions, we explain briefly this condition in Section 3.1. More specifically, the behavior of the normalizing function near the boundary of its domain was an- alyzed, considering that the Musielak–Orlicz function Φc does not satisfies the ∆2-condition. In [1] the authors found an example that has the same form of a ϕ-function, but does not satisfy all the properties of a ϕ-function. Our aim in this paper is to analyze the behavior of the normalizing function near the boundary of its domain, considering functions, as the one found in [1], which do not satisfy all the properties of the definition of a ϕ-function . 2 Preliminary Considerations In this section we provide an introduction to ϕ-families of probability distribu- tions. Let (T, Σ, µ) be a σ- finite, non-atomic measure space on which probability distributions are defined. In this space, T may be thought of as the set of real numbers R. These families are based on the replacement of the exponential func- tion by a ϕ-function ϕ: R → (0, ∞) that satisfies the following properties [7]: (a1) ϕ(·) is convex and injective; (a2) limu→−∞ ϕ(u) = 0 and limu→∞ ϕ(u) = ∞; (a3) There exists a measurable function u0 : T → (0, ∞) such that Z T ϕ(c + λu0)dµ < ∞, for all λ > 0, for every measurable function c: T → R for which R ϕ(c)dµ = 1. There are many examples of ϕ-functions that satisfy (a1)–(a3) [7]. In the definition of ϕ-function, the constraint R T ϕ(c)dµ = 1 can be replaced by R T ϕ(c)dµ < ∞ since this fact was shown in [1, Lemma 1]. Thus, condition (a3) can be rewritten as: (a3’) There exists a measurable function u0 : T → (0, ∞) such that Z T ϕ(c + λu0)dµ < ∞, for all λ > 0, for every measurable function c: T → R for which R T ϕ(c)dµ < ∞. Thus (a3) and (a3’) are equivalent. Also, there are functions that satisfy (a1)–(a2) but do not satisfy (a3’) and an example was given in [1, Example 2]: ϕ(u) = ( e(u+1)2 /2 , u ≥ 0, e(u+1/2) , u ≤ 0. (1) Clearly, limu→∞ ϕ(u) = ∞ and limu→−∞ ϕ(u) = 0. It was shown in [1] that for the function (1) there exists a measurable function c: T → R and another function u0, both functions were defined in [1], such that R T ϕ(c)dµ < ∞ but R T ϕ(c + u0)dµ = ∞. This function in (1) is a deformed exponential function up to a trivial multiplicative factor, as discussed in [3]. The ϕ-families of probability distributions were built based on the Musielak– Orlicz spaces [2]. Let ϕ be a ϕ-function. The Musielak–Orlicz function was de- fined in [7] by Φc(t, u) = ϕ(c(t) + u) − ϕ(c(t)), where c: T → R is a measurable function such that ϕ(c) is µ-integrable. Then we have, the Musielak-Orlicz space LΦc and the Musielak–Orlicz class L̃Φc , denoted by Lϕ c and L̃ϕ c , respectively. Let Kϕ c be the set of all the functions u ∈ Lϕ c such that ϕ(c + λu) is µ- integrable for every λ in a neighborhood of [0, 1]. We know that Kϕ c is an open set in Lϕ c [7, Lemma 2] and, for u ∈ Kϕ c , the function ϕ(c + u) is not necessarily in Pµ, so the normalizing function ψ: Kϕ c → R is introduced in order to make the density ϕ(c + u − ψ(u)u0) is in Pµ [7]. For any u ∈ Kϕ c , ψ(u) ∈ R is the unique function which ϕ(c + u − ψ(u)u0) is in Pµ [7, Proposition 3]. Let Bϕ c =  u ∈ Lϕ c : Z T uϕ′ +(t, c(t))dµ = 0  be a closed subspace of Lϕ c , thus for every u ∈ Bϕ c = Bϕ c ∩ Kϕ c , by the convexity of ϕ, one has ψ(u) ≥ 0 and ϕ(c + u − ψ(u)u0) ∈ Pµ. For each measurable function c: T → R such that p = ϕ(c) ∈ Pµ is associated a parametrization ϕc : Bϕ c → Fϕ c , given by ϕc(u) = ϕ(c + u − ψ(u)u0), where the operator ϕ acts on the set of real-value functions u: T → R given by ϕ(u)(t) = ϕ(u(t)) and the set Fϕ c = ϕc(Bϕ c ) ⊆ Pµ where Pµ = S {Fϕ c : ϕ(c) ∈ Pµ} and the map ϕc is a bijection from Bϕ c to Fϕ c . In the following section we will study the behavior of the normalizing function ψ near the boundary of Bϕ c . 3 The behavior of ψ near the boundary of Bϕ c Let us suppose that the condition (a3) (or the equivalent (a3’)) on ϕ-function definition is not fulfilled. In others words, it is possible to find functions e c: T → R such that R T ϕ(e c)dµ < ∞ but R T ϕ(e c + λu0)dµ = ∞ for some λ > 0. Now, for u being a function in ∂Bϕ c we want to know whether ψ(αu) converges to a finite value as α ↑ 1 or not. First, let us remember how the normalizing function ψ behaves near ∂Bϕ c , assuming that the condition (a3’) is satisfied and the Musielak–Orlicz function Φc does not satisfies the ∆2-condition [6]. 3.1 ∆2-Condition and the normalizing function The ∆2-condition of Musielak–Orlicz functions and ϕ-families of probability dis- tributions was studied in [6], where the behavior of the normalizing function ψ near the boundary of Bϕ c was discussed. Remember that the set Bϕ c = Kϕ c ∩ Bϕ c is open in Bϕ c , then a function u ∈ Bϕ c belongs to ∂Bϕ c , the boundary of Bϕ c , if and only if R T ϕ(c + λu)dµ < ∞ for all λ ∈ (0, 1) and R T ϕ(c + λu)dµ = ∞ for all λ > 1. We say that a Musielak–Orlicz function satisfies the ∆2-condition, if one can find a constant K > 0 and a non-negative function f ∈ L̃ϕ c such that Φ(t, 2u) ≤ KΦ(t, u), for all u ≥ f(t), and µ-a.e. t ∈ T. If the Musielak–Orlicz function Φc(u) = ϕ(c(t) + u) − ϕ(c(t)) satisfies the ∆2- condition, then R T ϕ(c + u)dµ < ∞ for all u ∈ Lϕ c and ∂Bϕ c is empty. Assuming that the Musielak–Orlicz function Φc does not satisfies the ∆2- condition, the boundary of Bϕ c is non-empty. Let u be a function in ∂Bϕ c , for α ∈ [0, 1). It was shown in [6, Proposition 6] that if R T ϕ(c + u)dµ < ∞ then the normalizing function ψu(α) = ψ(αu) → β, with β ∈ (0, ∞) as α ↑ 1, and if R T ϕ(c + u)dµ = ∞ then (ψu)′ +(α) → ∞ as α ↑ 1. Now, it follows our first result, which states that it is possible to show that ψ(αu) → ∞ as α ↑ 1, when R T ϕ(c + u)dµ = ∞, with u ∈ ∂Bϕ c , as in the following proposition: Proposition 1. For a function u ∈ ∂Bϕ c such that R T ϕ(c + u)dµ = ∞. Then ψ(αu) → ∞ as α ↑ 1. Proof. Suppose that, for some λ > 0, the function u satisfies ψ(αu) ≤ λ for all α ∈ [0, 1). Denote A = {u ≥ 0}. Observing that Z A ϕ(c+ αu − λu0)dµ ≤ Z T ϕ(c+ αu − λu0)dµ ≤ Z T ϕ(c+ αu − ψ(αu)u0)dµ = 1, we obtain that R A ϕ(c + u − λu0)dµ < ∞. In addition, it is clear that Z T \A ϕ(c + u − λu0)dµ ≤ Z T \A ϕ(c)dµ ≤ 1. As a result, we have R T ϕ(c + u − λu0)dµ < ∞. From the condition (a3’), it follows that R T ϕ(c + u)dµ < ∞, which is a contradiction. From this result we can investigate the behavior of ψ near the boundary of Bϕ c in terms of whether the condition (a3’) is satisfied or not. We will discuss about this in the following section. 3.2 The definition of ϕ-function and its consequences We know there are functions that satisfy conditions (a1)–(a2) in the definition of ϕ-function but do not satisfy (a3’) as seen in (1). In this section we discuss about the behavior of the normalizing function ψ near the boundary of Bϕ c in cases where the condition (a3’) is not satisfied. To begin with, let us prove that the condition (a3’) is equivalent to the existence of constants λ, α > 0 and a non-negative function f ∈ L̃ϕ c such that αΦc(t, u) ≤ Φc−λu0 (t, u), for all u > f(t). (2) For this we need the following lemma. Lemma 1. [2, Theorem 8.4] Let Ψ and Φ be finite-value Musielak–Orlicz func- tions. Then the inclusion L̃Φ ⊂ L̃Ψ is satisfied if and only if there exist α > 0 and a non-negative function f ∈ L̃Φ such that αΨ(t, u) ≤ Φ(t, u), for all u > f(t). Using the above lemma we can prove the equivalence between the condition (a3’) in the definition of ϕ-function and the inequality (2). Proposition 2. A measurable function u0 satisfies (a3’) in the definition of ϕ- function if and only if for some measurable function c: T → R such that ϕ(c) is µ-integrable, we can find constants λ, α > 0 and a non-negative function f ∈ L̃Φc such that αΦc(t, u) ≤ Φc−λu0 (t, u), for all u > f(t). (3) Proof. Suppose that u0 satisfies condition (a3’). Let c: T → R be any measurable function such that R T ϕ(c)dµ < ∞. As u is a measurable function with R T ϕ(c − λu0 + u)dµ < ∞ then Z T ϕ(c + u)dµ = Z T ϕ(c − λu0 + u + λu0)dµ < ∞. This result implies L̃Φc−λu0 ⊂ L̃Φc . Inequality (3) follows from Lemma 1. Now suppose that inequality (3) is satisfied. By Lemma 1 we have L̃Φc−λu0 ⊂ L̃Φc . Therefore, u ∈ L̃Φc implies u + λu0 ∈ L̃Φc−λu0 ⊂ L̃Φc . Or, equivalently, if u is a measurable function such that ϕ(c + u) is µ-integrable, then ϕ(c + u + λu0) is µ-integrable. As a result, we conclude that R T ϕ(c + u + λu0)dµ < ∞ for all λ > 0. Let e c: T → R be any measurable function satisfying R T ϕ(e c)dµ < ∞. Denote A = {e c > c}. Thus, for each λ > 0, it follows that Z T ϕ(e c+λu0)dµ = Z T ϕ(c+(e c−c)+λu0)dµ ≤ Z T ϕ(c+(e c−c)χA +λu0)dµ < ∞, which shows that u0 is stated in the definition of ϕ-functions. From this proposition we have that the condition (a3’) is satisfied if, and only if, there exists a measurable function u: T → R such that R T ϕ(c + u)dµ = ∞ but R T ϕ(c + u − λu0)dµ < ∞ for some λ > 0. For our main result we make use of the lemmas below. Lemma 2. [2, Lemma 8.3] Consider a non-atomic and σ-finite measure µ . If {un} is a sequence of finite-value, non-negative, measurable functions, and {αn} is a sequence of positive, real numbers, such that Z T undµ ≥ 2n αn, for all n ≥ 1, then an increasing sequence {ni} of natural numbers and a sequence {Ai} of pairwise disjoint, measurable sets can be found, such that Z Ai uni dµ = αni , for all i ≥ 1. For the next lemma we denote the functional IΦc = R T Φc(t, | u(t) |)dµ for any u ∈ L0 . Lemma 3. Consider c: T → [0, ∞) a measurable function such that R T ϕ(c)dµ < ∞. Suppose that, for each λ > 0, we cannot find α > 0 and f ∈ L̃Φc such that αΦc(t, u) ≤ Φc−λu0 (t, u), for all u > f(t). (4) Then a strictly decreasing sequence 0 < λn ↓ 0, and sequences {un} and {An}of finite-value, measurable functions, and pairwise disjoint, measurable sets, respec- tively, can be found such that IΦc (unχAn ) = 1, and IΦc−λnu0 (unχAn ) ≤ 2−n , for all n ≥ 1. (5) Proof. Let {λm} be a strictly decreasing sequence such that 0 < λm ↓ 0. Define the non-negative functions fm(t) = sup{u > 0 : 2−m Φc(t, u) > Φc−λmu0 (t, u)}, for all m ≥ 1, where we adopt the convention that sup ∅ = 0. Since (4) is not satisfied, we have that IΦc (fm) = ∞ for each m ≥ 1. For every rational number r > 0, define the measurable sets Am,r = {t ∈ T : 2−m Φc(t, r) > Φc−λmu0 (t, r)}, and the simple functions um,r = rχAm,r . For r = 0, set um,r = 0. Let {ri} be an enumeration of the non-negative rational numbers with r1 = 0. Define the non-negative, simple functions vm,k = max1≤i≤k um,ri , for each m, k ≥ 1. By continuity of Φc(t, ·) and Φc−λmu0 (t, ·), it follows that vm,k ↑ fm as k → ∞. In virtue of the Monotone Convergence Theorem, for each m ≥ 1, we can find some km ≥ 1 such that the function vm = vm,km satisfies IΦc (vm) ≥ 2m . Clearly, we have that Φc(t, vm(t)) < ∞ and 2−m Φc(t, vm(t)) ≥ Φc−λmu0 (t, vm(t)). By Lemma 2, there exist an increasing sequence {mn} of indices and a sequence {An} of pairwise disjoint, measurable sets such that IΦc (vmn χAn ) = 1. Taking λn = λm, un = vmn and An, we obtain (5). Finally, our main result follows. Proposition 3. Assuming that the condition (a3’) is not satisfied in the defi- nition of ϕ-function, then there exists u ∈ ∂Bϕ c such that R T ϕ(c+ u)dµ = ∞ but ψ(αu) → β, with β ∈ (0, ∞), as α ↑ 1. Proof. Let {λn}, {un} and {An} as in Lemma 3. Given any λ > 0, take n0 ≥ 1 such that λ ≥ λn for all n ≥ n0. Denote B = T \ S ∞ n=n0 An, then we define u = P∞ n=n0 unχAn . From (5), it follows that Z T ϕ(c + u − λu0)dµ = Z B ϕ(c − λu0)dµ + ∞ X n=n0 Z An ϕ((c − λu0) + un)dµ = Z B ϕ(c − λu0)dµ + ∞ X n=n0 Z An ϕ(c − λu0)dµ + IΦc−λu0 (unχAn )  ≤ Z T ϕ(c − λu0)dµ + ∞ X n=n0 2−n < ∞. Consequently, for α ∈ (0, 1), we can write Z T ϕ(c + αu)dµ = Z T ϕ  c + α(u − λu0) + (1 − α) αλ 1 − α u0  dµ ≤ α Z T ϕ(c + u − λu0)dµ + (1 − α) Z T ϕ  c + αλ 1 − α u0  dµ < ∞. On the other hand, for α ≥ 1, it follows that Z T ϕ(c + αu)dµ ≥ Z B ϕ(c)dµ + ∞ X n=n0 Z An ϕ(c + un)dµ ≥ Z B ϕ(c)dµ + ∞ X n=n0 Z An ϕ(c)dµ + IΦc (unχAn )  = Z T ϕ(c)dµ + ∞ X n=n0 1 = ∞. We can choose λ′ < 0 such that w = λ′ u0χB + ∞ X n=n0 unχAn satisfies R T wϕ′ +(c)dµ = 0. Clearly, R T ϕ(c + w)dµ = ∞, R T ϕ(c + αw)dµ < ∞ for α ∈ (0, 1) and R T ϕ(c + αw)dµ = ∞ for α > 1, that is, w ∈ ∂Bϕ c and R T ϕ(c + w − λu0)dµ < ∞ for some fixed λ > 0. Suppose that ψ(αw) ↑ ∞, then for all K > 0, there exists δ > 0 such that 0 < |α − 1| < δ implies that ψ(αw) > K. Let λ′′ > λ be such that R T ϕ(c+w−λ′′ u0)dµ < 1, taking K = λ′′ we have ϕ(c+αw−ψ(w)u0) < ϕ(c+αw{w>0} −λ′′ u0) < ϕ(c+w{w>0} −λ′′ u0), that is a µ-integrable function. Therefore by the Dominated Convergence Theorem we have lim α↑1 Z T ϕ(c + αw − λ′′ u0)dµ = Z T ϕ(c + w − λ′′ u0)dµ, then 1 = lim α↑1 Z T ϕ(c + αw − ψ(αw)u0)dµ ≤ lim α↑1 Z T ϕ(c + αw − λ′′ u0)dµ = Z T ϕ(c + w − λ′′ u0)dµ < 1, which is a contradiction. 4 Conclusions This paper focused on the behavior of the normalizing function ψ near the bound- ary of its domain. Assuming that the condition (a3’) is satisfied, it has been shown in [6] that for all measurable function u in the boundary of the normaliz- ing function domain such that E[ϕ(c + u)] = ∞, ψ(αu) converges to infinity as α approaches 1. Now, whereas that the condition (a3’) in ϕ-function definition is not satisfied, we found a measurable function w: T → R in the boundary of normalizing function domain such that E[ϕ(c + w)] = ∞ but ψ(αw) converges to a finite value as α approaches 1. We conclude that the condition (a3’) in the definition of ϕ-function affects the behavior of the normalizing function near the boundary of its domain. A perspective for future works is to investigate the be- havior of normalizing function considering that the ϕ-function is not necessarily injective. Acknowledgement The authors would like to thank CAPES and CNPq (Proc. 309055/2014-8) for partial funding of this research. References 1. David C. de Souza, Rui F. Vigelis, and Charles C. Cavalcante. Geometry induced by a generalization of rényi divergence. Entropy, 18(11):407, 2016. 2. Julian Musielak. Orlicz spaces and modular spaces, volume 1034 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1983. 3. Jan Naudts. Generalised thermostatistics. Springer-Verlag London, Ltd., London, 2011. 4. Giovanni Pistone, Maria Piera Rogantin, et al. The exponential statistical manifold: mean parameters, orthogonality and space transformations. Bernoulli, 5(4):721–760, 1999. 5. Giovanni Pistone and Carlo Sempi. An infinite-dimensional geometric structure on the space of all the probability measures equivalent to a given one. The annals of statistics, pages 1543–1561, 1995. 6. Rui F. Vigelis and Charles C. Cavalcante. The ∆2-condition and φ-families of prob- ability distributions. In Geometric science of information, volume 8085 of Lecture Notes in Comput. Sci., pages 729–736. Springer, Heidelberg, 2013. 7. Rui F. Vigelis and Charles C. Cavalcante. On ϕ-families of probability distributions. Journal of Theoretical Probability, 26(3):870–884, 2013. 8. Jun Zhang and Peter Hästö. Statistical manifold as an affine space: a functional equation approach. J. Math. Psych., 50(1):60–65, 2006.