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Letters
International Journal of Bifurcation and Chaos, Vol. 16, No. 1 (2006) 129–135
c World Scientific Publishing Company


GENERALIZATION OF THE MEAN-FIELDMETHOD FOR POWER-LAW DISTRIBUTIONSTARO TOYOIZUMIDepartment of Complexity Science and Engineering,
Graduate School of Frontier Sciences,
The University of Tokyo, 4-6-1 Komaba,
Meguro-ku, 153-8505 Tokyo, Japantaro@sat.t.u-tokyo.ac.jp
KAZUYUKI AIHARA
Department of Information and Systems,
Institute of Industrial Science,
The University of Tokyo, 4-6-1 Komaba,
Meguro-ku, 153-8505 Tokyo, Japan
ERATO Aihara Complexity Modelling Project,
JST, 3-23-5 Vehara, Shibuya-ku, 151-0064 Tokyo, Japan
aihara@sat.t.u-tokyo.ac.jp
Received October 31, 2004
Recently much attention has been paid to the nonextensive canonical distributions: the αfamilies. Such distributions have been found in many real-world systems such as fully developed
turbulence and financial markets. In this paper, a generalized mean-field method to approximate
the expectations of the α-families is proposed. We calculate the α -projection of a probability distribution to find that the computational complexity to approximate the expectations is greatly
reduced with a proper choice of the projection-index α . We apply this method to a simple
binary-state system and compare the results with direct numerical calculations.1
Keywords: Information geometry; generalized statistical mechanics; α-divergence.

p(x) = (1/Z)[1 − (1 − q)βH(x)]1/(1−q) . GSM
describes a large number of important real-world
phenomena with this single-parameter generalization such as self-gravitating systems [Taruya &Sakagami, 2002], long-range classical Hamiltoniansystems [Latora et al., 1998; Latora et al., 2001;
Latora & Tsallis, 2001], fully developed turbulence[Beck, 2001, 2002], financial markets [Ghashghaie
et al., 1996], and one-dimensional nonlinear maps
[Baldovin & Rovledo, 2002] (see [Abe & Okamoto,2001] for a summary). The more the importance of

1. Introduction
During the last decade, Generalized Statistical
Mechanics (GSM) has been intensively studied [Abe& Okamoto, 2001]. Adding one parameter, Tsallis[1988] proposed a generalizedversion of Shannon

entropy, Sq = −k[1 − x pq (x)]/(1 − q), where q
is the entropic index and p(x) is the microscopic
probability of a state x. As the limit of q → 1, theordinary Shannon entropy is derived. Maximizationof this entropy under an energy constraint yields
the GSM canonical distribution [Tsallis, 1988], i.e.
1This paper is a modification of the contribution in the proceedings of Shanghai International Symposium on Nonlinear Scienceand Applications 2003 (SNSA 2003).
129

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GSM is recognized, the greater the need for tools toanalyze the GSM canonical distribution. Because of
the correlations among the variables x of the GSM
canonical distribution, it is computationally hardto elucidate statistical properties of a large-size system. In this respect some methods, for example, themean-ﬁeld method and the variational method havebeen arranged for GSM [Plastino & Tsallis, 1993;
Lenzi et al., 1998; Mendes et al., 1999].
On the other hand, the mean-ﬁeld method is
now not only of interest to physicists but also
used in the ﬁelds of information theory [Kabashima& Saad, 1998] and machine learning [Peterson &Anderson, 1987; Opper & Winther, 2000]. That
is why the mean-ﬁeld method is intensively studied in the framework of the information geometry [Tanaka, 2000; Bhattacharyya & Keerthi, 2000].
Amari et al. proposed the α-projection of the
Boltzmann–Gibbs distribution as a generalizationof the mean-ﬁeld method [Amari et al., 2001]. In thispaper, however, we apply this method to power-law
distributions that are known as α-families in theﬁeld of information geometry [Amari & Nagaoka,
2000]. We show that the particular selection of a
projection enables us to approximate the expectations of a distribution with less computational
complexity compared with the exhaustive exact
calculation.
The main purpose of this paper is to computethe expectation η of an α-family:p(x; θ)

 Ñ




1


exp
θ ν fν (x) , Z(θ)
ν=1=
2/(1−α)Ñ
1−α  ν1
θ fν (x), Z(θ)2α=1

α = 1ν=0(1)where {fν }Ñν=0 is a set of linear independent functions of the state vector x = {x1 , . . . , xN }, θ ={θ ν }Ñν=0 is a coordinate system of the α-family, andZ(θ) is the normalization constant.
We assume f0 = 1, θ 0 = 2/(1 − α), andν[((1 − α)/2) Ñ
ν=0 θ fν (x)] > 0 for all x throughout this paper for simplicity. The expectations ofp(x; θ) are given asην ≡ Ep [fν ] =xfν (x)p(x; θ),

(2)


ν
for ν = 1, . . . , Ñ . When Ñν=1 θ fν (x) = −βH(x)and (1 − α)/2 = (1 − q), p(x; θ) of Eq. (1) is the
GSM canonical distribution.
Due to the nonlinear terms of {fν } in Eq. (1)for α = −1, it is computationally hard to calculate
η for large N systems. It takes O(N kN ) of computation for general α, where k is the number of
discrete states that xi can take. Therefore, we usually apply the mean-ﬁeld method to approximate
η. In the following, we propose a generalizationof the mean-ﬁeld method, that is the α-projection[Amari & Nagaoka, 2000; Amari et al., 2001] of the
α-family.

2. The α-Projections of Power-Law
Distributions
Let x = {xi |xi ∈ {1, . . . , k}, i = 1, . . . , N } be astate vector, S be the family of probability distributions on x, and M be the family of factorizable probability distributions on x. A probability
distribution on M is represented as p0 (x; h) =Nii Ni=1 p0 i (xi ; h ), where h = {h }i=1 is a coordinate system of M. In this section, we approximatep(x; θ)(∈ S) of Eq. (1) by its α -projection onto M.
We will show in the following that a proper
selection of the projection-index α can considerably reduce the computation of the α -projectionof p onto M [Toyoizumi & Aihara, 2003]. Thenwe will approximate the expectation η by η0 ≡{Ep0 [fi ]}N
i=1 . We will discuss the properties of theα -projection and explain that this method is a oneparameter generalization of the naive mean-ﬁeld
method.
Let us calculate the α -divergence Dα between
p and p0 to ﬁnd the α -projection of p onto M. Since
it is given by arg minp0 ∈M Dα (p||p0 ) (See Appendix
for properties of α -projection.), the α -divergence
between p and p0 is expressed as,
Dα (p||p0 ) =4p(1−α )/2 p(1+α )/2
1−2
1 − αx“4
ψ+ 1+α
− 1−α
Gα
2
2
1
−
e
=
1 − α 2”, (3)where ψ = log Z and2
1 + α
Gα ≡ −log Z + 2α −1(1−α )/2 (1+α )/2
p
p0
.
× log
x

(4)

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Generalization of the Mean-Field Method for Power-Law Distributions

Note that the second term of (4) is Rényi’s Gdivergence [Arndt, 2001]. It is easy to check thatGα is a monotonic increasing function of Dα
from Eq. (3), therefore arg minp0 ∈M Dα (p||p0 ) =arg minp0 ∈M Gα (p||p0 ). We minimize Gα insteadof Dα hereafter.
Then what α should we choose to approximate
η by η0 ? If M is a 1-autoparallel submanifold of
S, we have to calculate the (−1)-projection of p
for the exact expectations. However, the calculation of (−1)-projection is as computationally hardas that of direct calculation [Amari et al., 2001].
Thus we should choose a proper α taking accountof the computational complexity.
Because p is a distribution of the α-family,ν
it is represented as p(1−α)/2 = c1 Ñ
ν=0 θ fν + c2
with some constants c1 and c2 . Suppose we chooseα such that the α -divergence is a linear function of p(1−α)/2 , we do not have to deal withany nonlinear functions of {fν } to calculate thevalue of Dα (p||p0 ). Because of this reason, wechoose α = α here. In this case, Gα = −(4/(1 −ν00α2 )) log(((1 − α)/2) Ñ
ν=0 θ fν α ), with fν α ≡(1+α)/2 (x; h). If f (x) (ν = 0, 1, . . . , Ñ )νx fν (x)p0
are functions of l variables, it takes O(Ñ kl )steps to calculate Gα . In addition, we can also
calculate∂θ ν i fν 0α
∂h
4
∂Gα
ν

=−
(5)
i
2
∂h
1−α
θ ν fν 0αν

for i = 1, . . . , N at the same orders. One can easily
ﬁnd h that gives a local minimum of Gα by applying
an optimization algorithm.
In this way, this approximation greatly reduces
the number of operations for systems with large N ,while the exact calculation of η requires O(N kN )operations in general.

3. Application to a Binary-StateModel
In this section we calculate the α-projection of abinary-state distribution: the distribution of Eq. (1)with x = {xi |xi ∈ {+1, −1}, i = 1, . . . , N } and
N 
N i
Ñ
ν
ij
i=1
j>i θ xi xj + i=1 θ xi .
ν=1 θ fν (x) =Here we assume that minx [((1 − α)/2) ν θ ν fν ] =c(> 0). We approximate p by a p0 that minimizes
the α-divergence between them. Because x is abinary vector here, p0 is generally represented in

the following exponential form:
 N


1
p0 =
exp
hi xi .
Z0 (h)

131

(6)

i=1

Let us introduce the following expectations,



1+α
(α)
h
fν (x)p0 x;
η0 ν ≡
2
x





1+α j
1+α i


h
h
tan
, fν = xi xj
tanh


2
2
=



1+α i


h ,
fν = xi
tanh
2(7)for ν = 1, . . . , Ñ . Then we ﬁnd the gradient of
Eq. (5) to be


N

(α)
θ ij η0 j + θ i



2 
∂Gα

(α)
(α) j=1=
η0 − η0 i − g0 ii


Ñ
∂hi
1 − α i



ν (α)
θ η0 ν
ν=1

(8)
(α)

(α)

with g0 ii = [1 − (η0 i )2 ]. We employ the gradientdescent algorithm to search for a local minimum ofGα (Algorithm A in Table 1).
As the stationary condition of Algorithm A
for α → 1, we can derive the usual self-consistentequations ijof the inaive mean-ﬁeld method: ηi =tanh( j θ ηj + θ ) for i = 1, . . . , N . The reason
is that the naive mean-ﬁeld equation is derived
from the saddle point condition of the 1-divergence.
Therefore, in this case, the α-projection is a generalization of the naive mean-ﬁeld method for
α-families.

3.1. Numerical results
In this section we apply the method introducedin Sec. 3 to a system with small N and compare
Table 1.

Algorithm A.

1. Initialize h to small random values.
(α)(α)
2. Calculate η0 ν , g0 iν , and ∂Gα /∂hi .
3. Update h according toα,hinew = hiold − δ ∂G
∂hiwhere δ is a step size.
4. Return to step 2; stop after ﬁnite steps n∗ .

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the results with direct numerical calculations. We
also compare the results with another method that
could be applied: the mean-ﬁeld approximation of
the Callen identity [Sarmento, 1995].
Let us ﬁrst derive the mean-ﬁeld approximation of the Callen identity here. The single-site
Callen identity is ηi = Ep [(p(xi = 1, x\i ) − p(xi =−1, x\i ))/(p(xi = 1, x\i ) + p(xi = −1, x\i ))], wherex\i ≡ {x1 , . . . , xi−1 , xi+1 , . . . , xN }. Employing thenaive mean-ﬁeld approximation of the above equation, we obtain the self-consistent equations for η:ηi ≈p(xi = 1, η\i ) − p(xi = −1, η\i ),p(xi = 1, η\i ) + p(xi = −1, η\i )(9)for all i. We can compute the approximation of η
by calculating Eq. (9) repeatedly (Algorithm B in
Table 2).
Note that Algorithm B also has the same stationary condition as the naive mean-ﬁeld method
when α → 1. Therefore for α ≈ 1, their diﬀerences

Table 2.

Algorithm B.

1. Initialize η0 to small random values.
2. Update η0 using Eq. (9).
3. Return to step 2; stop after ﬁnite steps n∗∗ .

only come from the optimization algorithms that
we apply.
Figureof normalized error:
N 1 shows the average
|η
−
η
|/
|η
|) and its standard
Err ≡ ( N
i
0
i
i
i=1
i=1
deviation obtained from Algorithms A (cross) andB (square) for N = 10 and c = 1. {θ ij } and {θ i } are
generated from the Gauss distributions N (µ, 0.5)
and N (0.0, 1.0), respectively. We generate 100 samples here. Since it is generally diﬃcult to ﬁnd thestep size δ of Algorithm A, we repeat this algorithm
ﬁve times over diﬀerent δ = {1.0, 6.0, 11.0} andchoose the η0 that minimizes Dα after n∗ = 30 iterative steps. We carry out just one series (n∗∗ = 90
iterative steps) for Algorithm B. We show in Fig. 1that Algorithm A gives better results for every α

9

1.2

8

1

7
0.8

6
5

0.6

4

0.4

3

0.2

2
0

1
0

-0.2
-1 -0.8 -0.6 -0.4 -0.2 0

0.2 0.4 0.6 0.8

10.4-1 -0.8 -0.6 -0.4 -0.20

0.2 0.4 0.6 0.8

1-1 -0.8 -0.6 -0.4 -0.20

0.2 0.4 0.6 0.8

1

1.2

0.35

1

0.3
0.8

0.25
0.2

0.6

0.15

0.4

0.1
0.2

0.05
0

0
-1 -0.8 -0.6 -0.4 -0.2 0

Fig. 1.

0.2 0.4 0.6 0.8

1

The averaged normalized errors versus the mean of coupling coeﬃcient µ.

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Generalization of the Mean-Field Method for Power-Law Distributions

that are shown here. If α = −1, Dα have a uniqueminimum at which η = η0 holds [Amari et al.,2001]. Thus Algorithm A ﬁnds the exact expectations in this case, with a suﬃciently small step size δand many iterations. On the other hand, since the
α-divergence of Eq. (3) may take extremely large
values at p ≈ 0 (resp. p0 ≈ 0) for α > 1 (resp.
α < −1), it is possible that Algorithm A extracts
poor results in these cases.

133

0.19
0.18
0.17
0.16

MF

n=1

0.15

n=3
0.14

Direct
0.13
0.12

n=7

0.11

4. The Choice of Projections
What is important for the approximation of Sec. 2
is a proper choice of the projection. We have previously chosen the projection-index α that mostalleviate the computational cost to approximate
η. As we will see, there is a tradeoﬀ between thecomputational complexity and the accuracy of theapproximation.
If p(x; θ) is a distribution of the 1-family, i.e.
α = 1, the 1-projection is computationally theeasiest, while the (−1)-projection yields the exact
expectations [Amari et al., 2001]. As it is generally
diﬃcult to estimate to what extent the choice of the
projection-index aﬀects the approximation of η, we
take here the simplest model to compare several
projections.
Now we consider the α-family (|α| < 1) of
Eq. (1) with x = {xi |xi ∈ {+1, −1}, i = 1, 2}
and H(x) = θ 12 x1 x2 + θ 1 x1 + θ 2 x2 , where θ 12 =(2/(1 − α))J and θ 1 = θ 2 = (2/(1 − α))H.
First, a direct calculation gives the expectation
as in Eq. (13).

η=

0.1
0

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Fig. 2.

Approximations of η versus J.

Second, the naive mean-ﬁeld approximation ofCallen identity, i.e. Eq. (9), is calculated to be
Eq. (14).
Finally, for an integer n and α n = 1 − (1 − α)n,
we can also calculate the Gα n of Eq. (4) to be(α )Eq. (15), where η0 = tanh(((1 + α )/2)h), (c1 , H1 ,J1 ) = (1, H, J), andcn = cn−1 + 2HHn−1 + JJn−1 ,Hn = H(cn−1 + Jn−1 ) + (1 + J)Hn−1 ,Jn = Jn−1 + cn−1 J + 2HHn−1,

(10)
(11)
(12)

for n > 2; we choose the η0 that minimizes Eq. (15).
In Fig. 2 we show the expectations derived from
Eq. (13) (Direct), from Eq. (14) (MF), and from
Eq. (15) with n = {1, 3, 7} for α = 0.6 and H = 0.1.
Since α 5 = −1 in this case, the α n -projection gives
a good approximation when n ≈ 5.

[1 + 2H + J]2/(1−α) − [1 − 2H + J]2/(1−α),[1 + 2H + J]2/(1−α) + 2[1 − J]2/(1−α) + [1 − 2H + J]2/(1−α)(13)[1 + H(ηMF + 1) + JηMF ]2/(1−α) − [1 + H(ηMF − 1) − JηMF ]2/(1−α)
,
(14)
ηMF =
[1 + H(ηMF + 1) + JηMF ]2/(1−α) + [1 + H(ηMF − 1) − JηMF ]2/(1−α)

4
1 + α(α )(α ) 2h . (15)log(cn + 2Hn η0 + Jn (η0 ) ) − (1 + α ) log(2 cosh h) + 2 log 2 coshGα n = −1 − α2
2
When α ≈ 1, a larger n provides a better
approximation of η (as long as M is a 1-autoparallel
submanifold of S). Thus in this case there is a tradeoﬀ between the computational complexity and theprecision of the approximation. The method of theα -projection provides a parameter α that controls
this tradeoﬀ.

5. Conclusion
We have shown that it is possible to realize a generalization of the mean-ﬁeld method by calculating
the α-projection of power-law distributions. The
number of operations needed to approximate theFebruary 6, 2006

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expectations is greatly reduced with a proper choiceof the projection. We have applied this method toa simple binary-state α-family and compared themethod with the mean-ﬁeld approximation of theCallen identity. As a result of numerical calculations, the generalized mean-ﬁeld method provides
less errors for the expectations compared with theother method, especially when it is applied to αfamilies with α ≈ −1.
Although we have only considered factorizable
distributions to approximate the true distribution,
it is possible to obtain better approximation by considering a wider class of distributions. Since there
are a lot of useful techniques to deal with such structured distributions in the ﬁeld of information theory, this is an important problem for future study.
As α-families are attracting more and more
attention in ﬁelds such as fully developed turbulence, economics and self-organized criticality, it isimportant to study the applications of this method
to such complex systems.

Acknowledgments
The authors thank Hidetoshi Shimokawa, Kazuyuki
Tanaka, and Yoshiyuki Kabashima for beneﬁcial
discussions. This work is supported by the Research
Fellowships of the Japan Society for the Promotionof Science for Young Scientists and a Grant-in-Aid
for JSPS Fellows. This study is also partially supported by the Advanced and Innovational Research
program in Life Sciences and a Grant-in-Aid No.
17022017 on Priority Areas of the Ministry of Education, Culture, Sports, Science, and Technology,
from the Japanese Government.

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Amari, S. & Nagaoka, H. [2000] Methods of Information
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Amari, S., Ikeda, S. & Shimokawa, H. [2001] “Information geometry of α-projection in mean ﬁeld approximation,” in Advanced Mean Field Methods, eds.
Opper, M. & Saad, D. (MIT Press), pp. 241–257.
Arndt, C. [2001] Information Measures (SpringerVerlag).
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Beck, C. [2002] “Generalized statistical mechanics andfully developed turbulence,” Physica A 306, 189–198.
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Latora, V. & Tsallis, C. [2001] “Fingerprints of nonextensive thermodynamics in a long-range Hamiltoniansystem,” Physica A 305, 129–136.
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Appendix
A.1. Information Geometry
In this Appendix, we brieﬂy review the framework of information geometry. Information geometry describes the way to introduce a geometricalFebruary 6, 2006

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Generalization of the Mean-Field Method for Power-Law Distributions

structure into a space of probability distributions
once a divergence is given [Amari & Nagaoka, 2000].
Let S = {p(x; θ)} be an α-family. The α-divergence
between two probability distributions p = p(x; θ)and p = p(x; θ ) is deﬁned as41−p(1−α)/2 p(1+α)/2
Dα (p||p ) ≡1 − α2x(A.1)(p||p ) ≡ lim
for α = ±1, and D±1
α→±1 Dα (p||p )for α = ±1. Note that D−1 is equivalent to thewell-known Kullback–Leibler divergence. The nonnegative property of the α-divergence, Dα (p||p ) ≥0, is directly derived from Jensen’s inequality, where
the equality holds if and only if p = p . It is not symmetric except for α = 0 and satisﬁes Dα (p||p ) =D−α (p ||p). The α-divergence is a distance-like measure representing the diﬀerence between two probability distributions.
When p and p are close enough, i.e.

θ = θ + dθ, we can expand the α-divergence
betweenas Dα (p(θ)||p(θ + dθ)) = these pointsνλ(1/2) ν,λ gνλ (θ)dθ dθ + o((dθ)2 ), where the135

Fisher metric gνλ |p ≡ limp →p ∂ν ∂λ Dα (p||p ). θ isthe coordinate system of S and ∂ν ≡ (∂/∂θ ν ) is thenatural basis of θ. Note that [gνλ ] is a symmetricand positive deﬁnite matrix, invariant to the valueof α [Amari & Nagaoka, 2000]. We deﬁne the innerproduct of two basis at θ as ∂ν , ∂λ  ≡ gνλ (θ). Then
we can show the following two theorems (see [Amari& Nagaoka, 2000] for proofs):
Theorem 1. Let S be an α-family, p, q and r be

three points in S, γ1 be an α-geodesic connecting
p and q and γ2 be an (−α)-geodesic connecting q
and r in S. If the curves γ1 and γ2 are orthogonal at the intersection q, then the following
equality holds: Dα (p||r) = Dα (p||q) + Dα (q||r) −((1 − α2 )/4)Dα (p||q)Dα (q||r).
Theorem 2. Let S be an α-family, M a subman-

ifold of S, and p a point in S. A necessary andsuﬃcient condition for a point q ∈ M to be a stationary point of the function r → Dα (p||r) restricted
on M is that the α-geodesic connecting p and q to
be orthogonal to M at q. We call such q as an αprojection of p onto M.