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Estimators, escort probabilities, and phi-exponential families in statistical physics

The lower bound of Cramer and Rao is generalized to pairs of families of probability distributions, one of which is escort to the other. This bound is optimal for certain families, called phi-exponential in the paper. Their dual structure is explored.


Introduction
Aim of this paper is to translate some new results of statistical physics into the language of statistics. It is well-known that the exponential family of probability distribution functions (pdfs) plays a central role in statistical physics. When Gibbs introduced the canonical ensemble in 1901 [1] he postulated a distribution of energies E of the form where G is a normalization constant and where the control parameter β is the inverse temperature. Only recently [2], a proposal was made to replace (1) by a more general family of pdfs. The resulting domain of research is known under the name of Tsallis' thermostatistics. Some of the pdfs of Tsallis' thermostatistics are known in statistics under the name of Amari's α-family [3]. The latter have been introduced in the context of geometry of statistical manifolds [4]. The appearance of the same family of pdfs in both domains is not accidental. The apparent link between both domains is clarified in the present paper.
The new notion introduced in Tsallis' thermostatistics is that of pairs of families of pdfs, one of which is escort of the other [5]. Some basic concepts of statistics can be generalized by replacing at well-chosen places the pdf by its escort. In particular, we show in the next section how to generalize Fisher's information and, correspondingly, how to generalize the well-known lower bound of Cramer and Rao. Section 3 studies the statistical manifold of a family for which there exists an escort family satisfying the condition under which the generalized Cramer-Rao bound is optimal. This optimizing family has an affine geometry. Since this is usually the characteristic property of an exponential family a generalization of the latter seems indicated.
Section 4 shows how a strictly positive non-decreasing function φ of R + determines a function which shares some properties with the natural logarithm and therefore is called below a φ-logarithm. The inverse function is called the φ-exponential. In Section 5 it is used to define the φ-exponential family in the obvious way, by replacing the exponential function exp by the φ-exponential function. The standard exponential family is then recovered by the choice φ(x) = x, the α-family of Amari by φ(x) = x (1+α)/2 , the equilibrium pdfs of Tsallis' thermostatistics by the choice φ(x) = x q .
The next three sections are used to establish the dual parametrization of the φ-exponential family and to discover the role of entropy functionals. Section 6 introduces a divergence of the Bregman type. In Section 7 it is used to prove the existence of an information function (or entropy functional) which is maximized by the φ-exponential pdfs. Section 8 introduces dual parameters -in statistical physics these are energy and temperature. The paper ends with a short discussion in Section 9.
There have been already some attempts to study Tsallis' thermostatistics from a geometrical point of view. Trasarti-Battistoni [6] conjectured a deep connection between non-extensivity and geometry. He also gives general references to the use of geometric ideas in statistical physics. Several authors [7,8,9] have introduced a divergence belonging to Csiszár's class of f-divergences, which leads to a generalization of the Fisher information metric adapted to the context of Tsallis' thermostatistics. The relation with the present work is unclear since here the geometry is determined by a divergence of the Bregman type. Also the recent work of Abe [10] seems to be unrelated.

Estimators and escort pdfs
Fix a measure space Ω, µ. Let M 1 (µ) denote the convex set of all probability distribution functions (pdfs) p normalized w.r.t. µ Ω dµ(x) p(x) = 1. (2) Fix an open domain D of R n . Consider a family of pdfs p θ , parametrized with θ in D. The notation E θ will be used instead of E p θ . Simultaneously, a second family of pdfs (P θ ) θ∈D is considered. It is called the escort family. The notation F θ will be used instead of E P θ .
Recall that the Fisher information is given by A generalization, involving the two families of pdfs, is Clearly, the expression coincides with (4) if P θ = p θ . The following definition is a slight generalization of the usual definition of an unbiased estimator.
Definition 1 An estimator of the family (p θ ) θ∈D is a vector of random variables c k with the property that there exists a function F such that The function F will be called the scale function of the estimator.
The estimator is unbiased if F (θ) = 1 2 θ k θ k so that E θ c k = θ k . The well-known lower bound of Cramer and Rao can be written as for arbitrary u and v in R n . A similar lower bound, involving the information matrix g kl instead of Fisher's I kl , is now formulated.
Theorem 1 Let be given two families of pdfs (p θ ) θ∈D and (P θ ) θ∈D and corresponding expectations E θ and F θ . Let c be an estimator of (p θ ) θ∈D , with scale function F . Assume the regularity condition holds. Let g kl (θ) be the information matrix introduced before. Then, for all u and v in R n is .
The bound is optimal (in the sense that equality holds whenever u = v) if there exist a normalization function Z > 0 and a function G such that holds for all k in [1.
.m], for all θ ∈ D, and for µ-almost all x. In that case, c is an estimator of (P θ ) θ∈D with scale function G Proof Let From Schwartz's inequality follows The l.h.s. equals, using (8), The first factor of the r.h.s. equals The second factor of the r.h.s. equals This proves (9). Assume now that (10) holds. Combining it with the regularity condition (8) shows that c is an estimator for the escort family, with scaling function G. This makes it possible to write (10) as In this way one obtains On the other hand is Together with (18) this shows equality in (9) whenever u = v.
It is not investigated whether (10) is a necessary condition. For practical application of the lower bound one has to assume that c is also an estimator of the escort family (P θ ) θ∈D , with scale function G. The previous proposition shows that this is automatically the case when (10) is satisfied.
Example 1 Let µ be the Lebesgue measure restricted to [0, +∞) and let with θ > 0 and [u] + = max{u, 0}. The Fisher information I(θ) is divergent. Hence, the usual lower bound of Cramer and Rao is useless. Consider now the escort family Then one calculates This fixes the r.h.s. of the inequality (9). Let us estimate θ via its first moment, with c(x) = 3x. One has E θ c = θ, E θ c 2 = (3/2)θ 2 , F (θ) = θ 2 /2, Fc = 3θ and Fc 2 = 18θ 2 . Then (9) boils down to 3 Statistical manifold The well-known example of a family with optimal estimator is the exponential family One sees immediately that which is (17) with Z(θ) identically 1 and the escort pdf P θ equal to p θ . This example motivates also the geometric interpretation of (10), in the form (17), as a linear map between tangent planes. The score variables ∂ log p θ /∂θ k of the standard statistical manifold are replaced by the variables They are tangent vectors of the concave function G(θ) − θ l c l . The metric tensor of the latter function is a constant random variable. The geometry of the manifold of random variables (G(θ) − θ l c l ) θ∈D is transferred onto the family of pdfs (p θ ) θ∈D . Note that the score variables have vanishing expectation F θ . It is now obvious to define an inner product of random variables by Then one has Let g kl (θ) denote the inverse of g kl (θ) (assume it exists). Then a projection operator π θ onto the orthogonal complement of the tangent plane is defined by If (10) is satisfied, then This follows also immediately from That the derivatives of the score variables are linear combinations of the score variables and the constant random variable is usually the characteristic feature of the exponential family. This is a motivation to introduce a generalized notion of exponential family.

φ-logarithms and φ-exponentials
In the next section the notion of exponential family is generalized to a rather large class of families of pdfs. This is done by replacing the exponential function by some other function satisfying a minimal number of requirements. The latter function will be called a deformed exponential and will be denoted exp φ . This has the advantage that the resulting expressions look very familiar, resembling those of the exponential family. Fix an increasing function φ of [0, +∞), strictly positive on (0, +∞). It is used to define the φ-logarithm ln φ by Clearly, ln φ is a concave function which is negative on (0, 1) and positive on (1, +∞). The inverse of the function ln φ is denoted exp φ . It is defined on the range of ln φ . The definition can be extended to all of R by putting exp φ (u) = 0 if u is too small and exp φ = +∞ if u is too large. In case φ(u) = u for all u then ln φ coincides with the natural logarithm and exp φ coincides with the exponential function. Given φ, introduce a function ψ of R by Clearly is φ(u) = ψ(ln φ (u)) for all u > 0.
Proof First consider the case that [0, u) belongs to the range of ln φ . Then a substitution of integration variables v = ln φ (w) is possible. One finds, using dv/dw = 1/φ(w) and  Example 2 Let φ(u) = u q with q > 0. This function is increasing and strictly positive on (0, +∞). Hence, it defines a φ-logarithm which will be denoted ln q and is given by This deformed logarithm has been introduced in the context of nonextensive statistical physics in [11]. The inverse function is denoted exp q and is given by The function ψ is then given by Example 3 Let φ(x) = ⌈x⌉, the smallest integer not smaller than x. This piecewise constant function is increasing and strictly positive on (0, +∞). Hence, ln φ is piecewise linear. The function ψ is given by The φ-exponential exp φ is also piecewise linear and satisfies

The φ-exponential family
Let φ be given as in the previous section. Fix a measure space Ω, µ and a set of random variables c k , k = 1, · · · , n. The φ-exponential family of pdfs (p θ ) θ∈D is defined by The domain D is an open set of θ for which G(θ) exists such that (44) is properly normalized, i.e. p θ ∈ M 1 (µ). The distributions (44) are the equilibrium pdfs of generalized thermostatistics as introduced in [12,13].
Because φ(p θ (x)) cannot be zero for µ-almost all x one concludes that Z(θ) > 0 and that P θ is properly normalized. From the properties of the function exp φ follows immediately that This proves that (P θ ) θ∈D satisfies (10).

Example 2 continued
Let φ(u) = u q as in Example 2 above. The pdfs p θ are given by for θ in a suitable domain D. The escort probabilities are (assuming convergence of these integrals). The family (p θ ) θ∈D coincides with Amari's α-family [3], with α given by α = 2q − 1.

Divergences
Divergences of the Bregman type are needed for what follows. In the form given below they have been introduced in [14]. Fix a strictly positive increasing function φ of [0, +∞). Introduce D φ (p||p ′ ) ≥ 0 follows because ln φ is an increasing function. Also convexity in the first argument follows because ln φ is an increasing function. Let (p θ ) θ∈D be φ-exponential. Then infinitesimal variation of the divergence D φ (p||p ′ ) reproduces the metric tensor g kl (θ), up to a scalar function. Indeed, one has and Similar calculations give

Information content
In [15] the definition of deformed logarithm contains the additional condition that the integral converges. This condition is needed in the definition of entropy functional / information content based on the deformed logarithm. Introduce another strictly increasing positive function χ by The motivation for introducing this function comes from the fact that it satisfies the following property.
which is the desired result.
Define information content (also called entropy functional) I φ (p) of a pdf p in M 1 (µ) by whenever the integral converges. Using the lemma one verifies immediately that I φ (p) is a concave function of p. A short calculation gives This implies that and hence This relation links the divergence D φ (p||p ′ ) with the information function I φ (p).
The following result shows that the φ-exponential family is a conditional maximizer of I φ . It also shows that the scale function F is the Legendre transform of the information content I φ Theorem 2 Let (p θ ) θ∈D be φ-exponential, with estimator c and scale functions F and G. Then there exists a constant F 0 such that The minimum is attained for p = p θ . In particular, F (θ) is a convex function of θ and p θ maximizes I φ (p) under the constraint that Proof Let us first show that for any pdf p One has Hence, (70) becomes now But one has always D φ (p||p θ ) ≥ 0. Therefore, (73) follows. Next calculate, using the lemma, Because c is an estimator with scale function F one obtains Hence there exists a constant F 0 for which In combination with (73) this results in (71).
Without restriction one can assume F 0 = 0. In statistical physics the function F (θ) is the free energy as a function of temperature, up to some proportionality factor.
Example 4 Let φ(u) = u 2−q /q, with 0 < q < 2. This is of course only a reparametrization of example 2, which is done to recover expressions found in the literature. The deformed logarithm is given by One obtains χ(v) = v q and hence This is the entropy functional proposed by Tsallis [2] as a basis for nonextensive thermostatistics, and reported earlier in the literature by Havrda and Charvat [16] and by Daróczy [17]. The corresponding expression for the divergence is

Dual coordinates
Introduce dual coordinates Assume (10) holds. Then, one obtains from (19) To obtain the last line a φ-exponential family has been assumed. This relation implies Thes are the orthogonality relations between the two sets of coordinates θ and η. Next we derive the dual relation of (82).

Discussion
The present paper introduces generalized exponential families, and calls them φ-exponential because they depend on the choice of a strictly positive nondecreasing function φ of (0, +∞). Several properties, known to hold for the exponential family, can be generalized. The paper starts with a generalization of the well-known lower bound of Cramer and Rao, involving the concept of escort probability distributions. It is shown that the φ-exponential family optimizes this generalized lower bound. The metric tensor, which generalizes the Fisher information, depends on both the family of pdfs and the escort family, and determines the geometry of the statistical manifold.
In the final part of the paper deals with the dual structure of the statistical manifold. In statistical physics this duality maps mean energy onto temperature and entropy onto free energy. The dual structure is shown to exist in the general context of φ-exponential families.
Throughout the paper the number of parameters n has been assumed to be finite. A non-parametrized approach to statistical manifolds is found in [18]. The extension of the present work to this more abstract context has not been considered.