信息几何：近似随机性和独立性的应用探索

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"《Information Geometry, Near Randomness, and Near Independence》是一本专注于信息几何学在实际应用中的书籍，特别是涉及统计模型应用于受到Poisson过程、均匀过程或双变量过程中独立状态扰动的情况。作者Khadiga A. Arwini和Christopher T.J. Dodson通过信息几何提供了一个统一的微分几何框架，涵盖了众多应用领域，如蛋白质链中的氨基酸序列间距、密码学研究、通信和星系的聚类、宇宙空洞、随机纤维网络和随机多孔介质中的耦合空间统计以及量子混沌。书中还提供了数学统计、微分几何和概率密度函数信息几何的基础介绍章节。" 本书的核心概念是信息几何，它是一种利用微分几何方法来研究概率理论和统计模型的数学工具。在信息几何中，概率分布被看作是流形上的对象，而各种统计量则对应于该流形的几何特性。例如，熵可以视为一种度量分布复杂性的度量，而 Fisher 信息矩阵则反映了分布参数估计的精度。 "Near randomness" 和 "near independence" 是书中探讨的关键主题。近似随机性指的是某些数据集或过程在统计上与完全随机行为相似但不完全相同的情况。这种现象在许多自然和工程系统中都很常见，例如在蛋白质链的氨基酸序列中，虽然不是完全随机，但它们呈现出接近随机的模式。近似独立性则是指两个或多个变量之间看似独立但实际上可能存在微弱的关联，这在处理大数据集或复杂系统时尤为重要。书中的应用部分涉及了多种领域，包括： 1. 蛋白质链中的氨基酸序列间距：信息几何可以帮助分析蛋白质结构和功能，通过研究氨基酸出现的相对距离来揭示其生物功能。 2. 密码学研究：在密码学中，理解数据的近似随机性和独立性有助于设计更安全的加密算法。 3. 通信和星系的聚类：这些领域的研究可能涉及到信号在空间中的分布，信息几何可以提供一个框架来描述和分析这些分布的特征。 4. 宇宙空洞：在宇宙学中，信息几何可以用来描述星系在大尺度结构中的分布，特别是空洞和超星系团的形成。 5. 随机纤维网络和随机多孔介质：这些材料的性质取决于其内部结构的统计特性，信息几何可以帮助理解其复杂性和性能。 6. 量子混沌：在量子系统中，信息几何可以揭示混沌行为的统计特性，这对于理解和预测量子系统的动态行为至关重要。《Information Geometry, Near Randomness, and Near Independence》是一本将信息几何学深入到实际问题中的宝贵资源，对于科学家和学生来说，无论是从事材料科学、信息科学还是物理、生物学等领域的研究，都能从中获益。通过学习和应用书中的理论和方法，读者能够更好地理解和处理现实世界中的复杂数据和随机现象。

1.2 Probability Density Functions for Continuous Variables 7

Again with slight abuse of notation, we call x the mean and the variance is

the mean square deviation

= (x − x)

= x

− (x)

Its square root is the standard deviation σ



− (x)

and the ratio

/x is the coeﬃcient of variation, of the random variable x. Some inequalities

for the probability of a random variable exceeding a given value are worth

mentioning.

Markov’s Inequality: If x is a nonnegative random variable with probabil-

ity density function f then for all a>0, the probability that x>ais



∞

f ≤

¯x

. (1.29)

Chebyshev’s Inequality: If x is a random variable having probability den-

sity function f with zero mean and ﬁnite variance σ

, then for all a>0,

the probability that x>ais



∞

f ≤

+ a

. (1.30)

Bienaym´e-Chebyshev’s Inequality: If x is a random variable having

probability density function f and u is a nonnegative non-decreasing

function on (0, ∞), then for all a>0 the probability that |x| >ais

1 −



−a

f ≤

¯u

u(a)

. (1.31)

The cumulative distribution function (cdf) of a nonnegative random variable x

with probability density function f is the function deﬁned by

F :[0, ∞) → [0, 1] : x →



f(t) dt. (1.32)

It is easily seen that if we wish to change from random variable x with

density function f to a new random variable ξ when x is given as an invertible

function of ξ, then the probability density function for ξ is represented by

g(ξ)=f (x(ξ))



dξ



. (1.33)

If independent real random variables x and y have probability density func-

tions f,g respectively, then the probability density function h of their sum

z = x + y is given by

h(z)=



∞

−∞

f(x) g(z − x) dx (1.34)

1.3 Joint Probability Density Functions 9

and it is not diﬃcult to show that the probability density function for the

Poisson inter-event (or inter-incident) distance t on [0, ∞) is an exponential

probability density function (1.39) given by

f :[0, ∞) → [0, ∞):t →

−t/τ

where τ is the mean number of events per unit interval. Thus, the occurrence

of an exponential distribution has associated with it a complementary Poisson

distribution, so the exponential distribution provides for continuous variables

an identiﬁer for Poisson processes. Correspondingly, departures from an ex-

ponential distribution correspond to departures from a Poisson process. We

shall see below in §1.4.1 that in rather a strict sense the gamma distribution

generalises the exponential distribution.

1.2.3 Gaussian, or Normal Distribution

This has real random variable x with mean µ and variance σ

and the familiar

bell-shaped probability density function given by

f(x)=

√

2πσ

−(x−µ)

2σ

. (1.42)

The Gaussian distribution has the following uniqueness property: For indepen-

dent random variables x

,...,x

with a common continuous probability

density function f, having independence of the sample mean ¯x and sample

standard deviation S is equivalent to f being a Gaussian distribution [110].

The Central Limit Theorem states that for independent and identically

distributed real random variables x

each having mean µ and variance σ

, the

random variable

w =

+ x

+ ...+ x

) − nµ

√

nσ

(1.43)

tends as n →∞to a Gaussian random variable with mean zero and unit

variance.

1.3 Joint Probability Density Functions

Let f be a probability density function, deﬁned on R

(or some subset thereof).

This is an important case since here we have two variables, X, Y, say, and we

can extract certain features of how they interact. In particular, we deﬁne their

respective mean values and their covariance, σ

x =



∞

−∞



∞

−∞

xf(x, y) dxdy (1.44)

y =



∞

−∞



∞

−∞

yf(x, y) dxdy (1.45)



∞

−∞



∞

−∞

xy f(x, y) dxdy − x y = xy − x y. (1.46)

1.4 Information Theory 11

with

W = −

2(σ

− σ

)



(x − µ

)

− 2 σ

(x − µ

)(y − µ

)+σ

(y − µ

)



where

−∞ <x

< ∞, −∞ <µ

<µ

< ∞, 0 <σ

,σ

< ∞.

This contains the ﬁve parameters (µ

,µ

,σ

)=(ξ

,ξ

) ∈ Θ.

So we have a ﬁve-dimensional parameter space Θ.

1.4 Information Theory

Information theory owes its origin in the 1940s to Shannon [186], whose in-

terest was in modelling the transfer of information stored in the form of

binary on-oﬀ devices, the basic unit of information being one bit: 0 or 1.

The theory provided a representation for the corruption by random electronic

noise of transferred information streams, and for quantifying the eﬀective-

ness of error-correcting algorithms by the incorporation of redundancy in the

transfer process. His concept of information theoretic entropy in communica-

tion theory owed its origins to thermodynamics but its eﬀectiveness in general

information systems has been far reaching. Information theory worked out by

the communication theorists, and entropy in particular, were important in

providing a conceptual and mathematical framework for the development of

chaos theory [93]. There the need was to model the dynamics of adding small

extrinsic noise to otherwise deterministic systems. In physical theory, entropy

provides the uni-directional ‘arrow of time’ by measuring the disorder in an

irreversible system [164]. Intuitively, we can see how the entropy of a state

modelled by a point in a space of probability density functions would be ex-

pected to be maximized at a density function that represented as nearly as

possible total disorder, colloquially, randomness.

Shannon [186] considered an information source that generates symbols

from a ﬁnite set {x

|i =1, 2, ···n} and transmits them as a stationary sto-

chastic process. He deﬁned the ‘entropy’ function for the process in terms of

the probabilities {p

|i =1, 2, ···n} for generation of the diﬀerent symbols:

S = −

i=n



i=1

log(p

). (1.53)

This entropy (1.53) is essentially the same as that of Gibbs and Boltzmann in

statistical mechanics but here it is viewed as a measure of the ‘uncertainty’ in

the process; for example S is greater than or equal to the entropy conditioned

by the knowledge of a second random variable. If the above symbols are gener-

ated mutually independently, then S is a measure of the amount of information

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