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Yu.I. Bogdanov LANL Report quant-ph/0303014

 

(n + m)= λ1 2 E .

(2.9)

The sample mean energy E (i.e., the sum of the mean potential energy that can be measured in the coordinate space and the mean kinetic energy measured in the momentum space) was used as the estimator of the mean energy in numerical simulations below.

Now, let us turn to the study of statistical fluctuations of a state vector in the problem under consideration. We restrict our consideration to the energy representation in the case when the expansion basis is formed by stationary energy states (that are assumed to be nondegenerate).

Additional condition (2.3) related to the conservation of energy results in the following relationship between the components

δ

 

=s1

(Ejcjδcj +Ejδcjcj )=0.

 

E

(2.10)

 

 

j=0

 

 

It turns out that both parts of the equality can be reduced to zero independently if one assumes that a state vector to be estimated involves a time uncertainty, i.e., may differ from the true one by a small time translation. The possibility of such a translation is related to the time-energy uncertainty relation.

The well-known expansion of the psi function in terms of stationary energy states, in view

of time dependence, has the form ( h =1)

 

ψ(x)= cj exp(iEj (t t0 ))ϕj (x)=

 

j

 

= cj exp(iEjt0 )exp(iEjt)ϕj (x)

(2.11)

 

j

 

In the case of estimating the state vector up to translation in time, the transformation

 

cj = cj exp(iEjt0 )

(2.12)

related to arbitrariness of zero-time reference t0 may be used to fit the estimated state vector to the

true one.

The corresponding infinitesimal time translation leads us to the following variation of a state

vector:

 

δcj =it0Ejcj .

(2.13)

Let δc be any variation meeting both the normalization condition and the law of energy

conservation. Then, from (1.15) and (2.3) it follows that

 

(δcj )cj =iε1 ,

(2.14)

j

 

(δcj )Ejcj =iε2 ,

(2.15)

j

 

where ε1 and ε2 are arbitrary small real numbers.

 

In analogy with Sec.1, we divide the total variation δc into

unavoidable physical

fluctuation δ2c and variations caused by the gauge and time invariances:

 

δcj =iαcj +it0Ejcj 2cj .

(2.16)

11

Yu.I. Bogdanov LANL Report quant-ph/0303014

We will separate out the physical variation δ2c , so that it fulfills the conditions (2.14) and

(2.15) with a zero right part. It is possible if the transformation parameters α and

t0 satisfy the

following set of linear equations:

 

 

α +

 

 

 

 

 

 

 

 

Et

0

1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

.

(2.17)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Eα + E

2

t0 2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

The determinant of (2.17) is the energy variance

 

 

 

 

 

 

σE2 =

E2

 

2 .

 

 

E

 

(2.18)

We assume that the energy variance is a positive number. Then, there exists a unique solution of the set (2.17). If the energy dissipation is equal to zero, the state vector has the only nonzero component. In this case, the gauge transformation and time translation are dependent, since they are reduced to a simple phase shift.

In full analogy with the reasoning on the gauge invariance, one can show that in view of both the gauge invariance and time homogeneity, the transformation satisfying (2.17) provides minimization of the total variance of the variations (sum of squares of the components absolute values). Thus, one may infer that physical fluctuations are minimum possible fluctuations compatible with the conservation of norm and energy.

Let us make an additional remark about the time translation introduced above. We will estimate the deviation between the estimated and exact state vectors by the difference between their scalar product and unity. Then, one can state that the estimated state vector is closest to the true one specified in the time instant that differs from the “true” one by a random quantity. In other words, a quantum ensemble can be considered as a time meter, statistical clock, measuring time within the accuracy up to the statistical fluctuation asymptotically tending to zero with increasing number of particles of the system. Time measurement implies the comparison between the readings of real and reference systems. The dynamics of the state vector of a reference system corresponds to an infinite ensemble and is determined by the solution of the Schrödinger equation. The estimated state vector of a real finite system is compared with the “world-line” of an ideal vector in the Hilbert space, and the time instant when the ideal vector is closest to the real one is considered as the reading of the statistical clock. Note that ordinary clock operates in the similar way: their readings correspond to the situation when the scalar product of the real vector of a clock hand and the reference one making one complete revolution per hour reaches maximum value.

 

Assuming that the total variations are reduced to the physical ones, we assume hereafter that

 

(δcj )cj = 0,

(2.19)

 

 

j

 

 

 

 

 

(δcj )Ejcj = 0.

(2.20)

 

 

j

 

 

 

 

 

The relationships found yield (in analogy with Sec.1) the conditions for the covariance

 

 

 

 

 

matrix

Σ

ij

=

δc δc

 

 

 

 

i j :

 

 

(Σijcj )= 0 ,

(2.21)

 

 

j

 

 

 

 

 

(Σij Ejcj )= 0 .

(2.22)

 

 

j

 

 

 

 

 

Consider the unitary matrix U +

with the following two rows (zero and first):

12

Yu.I. Bogdanov LANL Report quant-ph/0303014

(U + )

 

= c

 

 

(2.23)

0 j

 

j ,

 

 

(U + )

 

 

(Ej

 

)cj

 

 

 

=

E

j =0,1,...,s 1

.

(2.24)

 

 

 

 

1 j

 

 

σE

 

 

 

 

 

 

 

This matrix determines the transition to principle components of the variation

 

Uij+δcj

fi .

 

 

(2.25)

According to (2.19) and (2.20), we have δf0 f1 = 0 identically in new variables so

that there remain only s 2 independent degrees of freedom. The inverse transformation is

Uδf c .

 

 

 

 

(2.26)

On account of the fact that δf0 f1 = 0, one may drop two columns (zero and first) in

the U matrix turning it into the factor loadings matrix L

 

Lijδf j ci

i = 0,1,...,s 1; j = 2,3,...,s 1.

(2.27)

The L matrix has s rows and

s 2 columns. Therefore, it provides the transition from s 2

independent variation principle components to s components of the initial variation.

 

In principal components, the Fisher information matrix and covariance matrix are given by

Iijf

= (n +m)δij ,

 

 

 

(2.28)

 

 

 

 

(n +m)

 

 

 

 

 

 

 

 

Σijf

fiδf j =

1

 

 

δij .

(2.29)

 

 

 

In order to find the covariance matrix for the state vector components, we will take into

account the fact that the factor loadings matrix

 

 

L differs form the unitary matrix

absence of two aforementioned columns, and hence,

 

 

L L+ =δ −c c

(Ei

 

)(Ej

 

)

 

c c

 

E

E

,

 

 

 

σ2

 

 

 

 

 

 

 

 

 

ik

kj

 

ij

 

 

i

j

 

 

 

 

 

 

 

 

 

 

 

 

i

j

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

E

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

L L+

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Σ =δc δc = L L δf δf

=

ik kj

 

 

 

n

+m .

 

 

 

ij

 

 

i

 

j

 

ik jr

k

r

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Finally, the covariance matrix in the energy representation takes the form

 

 

 

 

 

 

 

 

 

 

 

(E

 

)(E

 

 

 

)

 

 

 

 

1

 

 

 

E

j

E

Σ

 

=

 

 

δ

 

c c 1+

 

 

i

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

σ 2

 

 

 

 

 

 

 

 

 

ij

 

(n +m)

ij

i j

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

E

 

 

 

 

 

 

 

 

.

i, j = 0,1,...,s 1

U by the

(2.30)

(2.31)

(2.32)

It is easily verified that this matrix satisfies the conditions (2.21) and (2.22) resulting from the conservation of norm and energy.

The mean square fluctuation of the psi function is

 

dx =

 

 

 

 

s 2

 

δψδψ

δciϕi (x)δcjϕj (x)dx ciδci =Tr(Σ)=

. (2.33)

n +m

 

 

 

 

 

 

 

 

13

s =100

Yu.I. Bogdanov LANL Report

quant-ph/0303014

 

The estimation of optimal number of harmonics in the Fourier series, similar to that in Sec.6

of the Paper.1, has the form

 

 

 

 

s

= r +1 rf (n + m)

,

(2.34)

opt

 

 

where the parameters r and

f

 

determine the asymptotics for the sum of squares of residuals:

 

f

 

 

 

Q(s)=

 

ci

 

2 =

 

.

(2.35)

 

 

 

 

 

r

 

 

i=s

s

 

 

 

The norm existence implies only that r > 0 . In the case of statistical ensemble of harmonic

oscillators with existing energy, r >1. If the energy variance is defined as well, r > 2.

Figure 4 shows how the constraint on energy decreases high-energy noise. The momentumspace density estimator disregarding the constraint on energy (upper plot) is compared to that accounting for the constraint.

Fig. 4 (a) An estimator without the constraint on energy;

(b) An estimator with the constraint on energy.

P(p)

0,6 a)

s=3

0,5

0,4

0,3

0,2

s=100

0,1

 

0,0

-2

-1

0

1

2

p

P(p)

0,6

b)

0,5

s=3

 

0,4

 

0,3

s=100

0,2

0,1

0,0

-2

-1

0

1

2

p

The sample of the size of n = m =50 ( n +m =100) was taken from the statistical ensemble of harmonic oscillators. The state vector of the ensemble had three nonzero components ( s =3).

Figure 4 shows the calculation results in bases involving s =3 and functions, respectively. In the latter case, the number of basis functions coincided with the total sample size. Figure 4 shows that in the case when the constraint on energy was taken into account, the 97 additional noise components influenced the result much weaker than in the case without the constraint.

14

Yu.I. Bogdanov LANL Report quant-ph/0303014

3. Spin State Reconstruction

The aim of this section is to show the application of the approach developed above to reconstruct spin states (by an example of the ensemble of spin-1/2 particles).

We will look for the state vector in the spinor form

 

1

 

0

c

 

 

 

 

 

 

 

 

1

 

 

 

 

ψ =c1

0 +c2

1

= c

 

 

(3.1)

 

 

 

 

 

2

 

 

 

 

The spin projection operator to direction nr is

 

 

P(s 1/ 2)=

1 (1±σrnr)

 

(3.2)

 

n

r

2

 

 

 

 

 

 

 

 

 

 

We will set n by the spherical angles

 

 

nr = (nx , ny , nz )= (sinθ cosϕ, sinθ sin ϕ, cosθ )

(3.3)

r

The probabilities

for

a particle to

have positive and negative projections along the n

direction are

 

 

 

 

 

 

 

 

P (nr)= 1 ψ P(s =+1/ 2)ψ = 1 ψ 1rnrψ

(3.4)

 

+

2

 

n

 

 

2

 

 

 

 

 

 

 

 

P (nr)= 1 ψ P(s =−1/ 2)ψ =

1 ψ 1−σrnrψ

(3.5)

 

2

 

n

 

 

2

 

 

 

 

 

 

 

 

respectively.

Direct calculation yields the following expressions for the probabilities under consideration:

P+(nr)=P+(θ,ϕ)= 12[(1+cosθ)c1*c1 +sinθ eiϕc1*c2 +sinθ eiϕc2*c1 +(1cosθ)c2*c2 ] (3.6)

P(nr)= P(θ,ϕ)= 12[(1cosθ)c1*c1 sinθ eiϕc1*c2 sinθ eiϕc2*c1 +(1+cosθ)c2*c2 ] (3.7)

 

These probabilities evidently satisfy the condition

 

 

P+(θ,ϕ)+P(θ,ϕ)=1

(3.8)

 

The likelihood function is given by the following product over all the directions involved in

measurements:

 

 

 

L =(P+(nr))N+(nr)(P(nr))N(nr)

(3.9)

 

r

 

Here, N+(nr) andn

N(nr) are the numbers of spins with positive and negative projections along

the nr

direction. In order to reconstruct the state vector of a particle, one

has to conduct

measurements at least in three noncomplanar (linearly independent) directions.

 

The total number of measurements is

 

 

N =N(nr)=(N+(nr)+N(nr))

(3.10)

 

nv

nv

 

15

Yu.I. Bogdanov LANL Report quant-ph/0303014

In complete analogy to (1.8), we will find the likelihood equation represented by the set of two equations in two unknown complex numbers c1 and c2 .

1

 

 

r

 

 

 

iϕ

 

r

 

 

 

 

 

iϕ

 

 

 

 

N+ (rn)

[(1

+ cosθ )c1 +sinθe

 

c2 ]+

N(rn)

[(1

cosθ )c1 sinθe

 

 

c2

]

= c1

(3.11)

N

 

 

 

nr

 

P+ (n)

 

 

2

 

 

P(n)

 

 

 

2

 

 

 

 

 

 

 

1

 

 

r

 

iϕ

 

 

 

r

 

iϕ

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

N+ (rn)

[ sinθe

 

c1 + (1cosθ )c2 ]+

N(rn)

[sinθe

 

c1 + (1+ cosθ )c2 ]

= c2

(3.12)

N

 

 

nr

 

P+ (n)

 

 

2

 

 

P(n)

P (nr)

2

P (nr)

 

 

 

 

 

 

 

This system is nonlinear, since the probabilities

and

depend on the unknown

 

 

+

 

 

 

amplitudes c1 and c2 .

This system can be easily solved by the method of iterations. The resultant estimated state vector will differ from the true state vector by an asymptotically small random number (the squared absolute value of the scalar product of the true and estimated vectors is close to unity). The corresponding asymptotical formula has the form

 

 

 

+

(0)

 

2

~2

χ42j

 

 

 

 

N 1

 

c c

 

 

 

2 j =

 

(3.13)

 

 

 

 

 

 

 

 

 

 

 

 

2

 

 

 

 

 

 

 

Here, χ42j

is the random number with the chi-square distribution of 4 j degrees of freedom,

and j is the particle spin ( j =1/ 2 in our case).

The validity of the formula (3.13) is illustrated with Fig. 5. In this figure, the results of 100

numerical experiments are presented. In each experiment, the spin j =1/ 2 of a pure ensemble has been measured along 200 random space directions by 50 particles in each direction, i.e.,

N =200 50=10000. In this case, the left side of (3.13) is the half of a random variable with the chi-square distribution of 4j =2 degrees of freedom.

16

Yu.I. Bogdanov LANL Report quant-ph/0303014

Incoherent mixture is described in the framework of the density matrix. Two samples are referred to as mutually incoherent if the squared absolute value of the scalar product of their state vector estimates is statistically significantly different from unity. The quantitative criterion is

 

 

N N

 

 

 

+(1) (2)

 

2

 

 

2

 

χα2,4 j

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

2

 

c

 

c

 

 

~

 

 

 

 

 

 

 

 

 

 

1

 

 

 

 

=

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

N +N

 

 

 

 

 

 

 

 

α,2 j

 

2

 

 

 

(3.14)

 

1

2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Here,

N and

N

are the sample sizes, and

χ2

 

is the quantile corresponding to the

 

 

 

1

 

 

 

2

 

 

 

 

 

 

 

 

 

α,2 j

 

significance level α for the chi-square distribution of

2 j

 

degrees of freedom. The significance

level

α describes the probability of error of the first kind, i.e., the probability to find samples

inhomogeneous while they are homogeneous (described by the same psi function).

 

Let us

outline

the

process

of

reconstructing states

with arbitrary spin. Let ψm(j) be the

amplitude of the probability to have the projection m along the z axis in the initial coordinate

frame (these

are

 

the

quantities to be estimated

by the results of

measurements),

 

 

 

 

 

~

 

 

 

 

 

 

 

 

 

 

 

(j)

be the corresponding quantities in the rotated coordinate frame.

m=( j, j 1,...,j). Let ψm

 

 

 

 

 

 

 

 

(j)

 

2

 

 

 

 

 

 

 

 

 

 

 

The probability to get the value m in measurement along the zaxis is

~

 

.

 

ψm

 

 

Both rotated and initial amplitudes are related to each other by the unitary transformation

 

(j)

 

*(j)

(j)

 

 

 

 

 

 

~

=D

ψ

 

 

 

 

 

 

ψ

m

 

 

 

 

 

(3.15)

 

 

mm

m

 

D(j)(α,β,γ)

The matrix

D(j)

is a function of the Euler angles

, where the angles

 

mm

mm

α and β coincide with the spherical angles of the zaxis with respect to the initial coordinate frame xyz, so that α =ϕ and β =θ . The angle γ corresponds to the additional rotation of the coordinate frame with respect to the zaxis (this rotation is insignificant in measuring the spin

projection along the zaxis, and it can be set γ =0). The matrix Dm(jm)is described in detail in

[14]. Note that our transformation matrix in (3.15) corresponds to the inverse transformation with respect to that considered in [14].

 

The likelihood equation in this case has the form

 

 

 

 

1

 

N (ϕ,θ)D(j)

(ϕ,θ)

(j)

 

 

 

 

 

 

 

 

m

*(j)

 

 

m

 

 

 

 

 

 

 

 

 

 

 

m m

 

 

 

 

 

 

 

 

N

′ ϕ θ

 

 

ψ

 

 

 

 

 

,

 

(3.16)

 

 

 

m , ,

 

 

~

 

 

 

 

 

 

 

 

 

(ϕ,θ)

 

m

 

 

 

 

 

 

where

N

 

 

 

 

 

 

 

 

 

m

 

 

is the number of spins with the projection m along the

z

axis with

direction determined by the spherical angles ϕ and θ, and N is the total number of measured spins.

4. Mixture Division

There are two different methods for constructing the likelihood function for a mixture resulting in two different ways to estimate the density (density matrix). In the first method (widely

17

Yu.I. Bogdanov LANL Report quant-ph/0303014

used in problems of estimating quantum states [7-10]), the likelihood function for the mixture is constructed regardless a mixed (inhomogeneous) data structure (structureless approach). In this case, for the two-component mixture, we have

L0

= n

p(xi

 

c(1),c(2), f1 , f2 ),

(4.1)

 

 

i=1

 

 

 

 

where the mixture density is given by

 

p(x)= f1 p1 (x)+ f2 p2 (x);

(4.2)

p1 (x) and

p2 (x) are the densities of the mixture components; f1

and f2 , their weights.

The normalization condition is

 

f1 + f2 =1

(4.3)

Here, it is assumed that all the points of the mixed sample

xi ( i =1,2,..., n ) are taken from

the same

distribution p(x)= f1 p1 (x)+ f2 p2 (x). In other

words, the mixture of two

inhomogeneous samples (i.e., taken from two different distributions) is treated as a homogeneous sample taken from an averaged distribution.

The second approach, which seems to be more physically adequate, is based on the notion of a mixture as an inhomogeneous population (component approach). This implies that the mixed

sample is considered as an

inhomogeneous population with n1 f1n points taken from the

distribution with the density

p1 (x) and the other n2 f2 n points, from the distribution with the

density p2 (x).

 

This approach has also formal advantages compared to the first one: it provides higher value of the likelihood function (and hence, higher value of information); besides that, basic theory structures, such as the Fisher information matrix and covariance matrix, take a block form. Thus, the problem is reduced to the division of an inhomogeneous (mixed) population into homogeneous (pure) sub-populations.

In view of the mixed data structure, the likelihood function in the component approach is given by

L1 = n1

p1 (xi

 

c(1))n

p2 (xi

 

c(2)).

(4.4)

 

 

i=1

 

 

i=n1 +1

 

 

 

 

Two different cases are possible:

1- population is divided into components a priory from the physical considerations (for instance, n1 values are taken from one source and n2 , from another source)

2- dividing the population into components have to be done on the basis of data itself without any information about sources (“blindly”)

The first case does not present difficulties, since it is reduced to analyzing homogeneous components in turn. The case when prior information about the sources is lacking requires additional considerations. In order to divide the mixture in this case, we will employ the so-called

randomized (mixed) strategy. In this approach, we will consider the observation xi to be taken

from the first distribution with the probability

 

 

 

 

 

f1 p1 (xi )

 

 

 

 

; and from the second one,

 

f

p

(x

i

)+ f

2

p

2

(x

i

)

 

 

 

f2 p2 (xi )

 

 

 

 

 

 

 

 

 

 

 

1 1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

. Having divided the sample into two population, we will find new estimates

 

f

p

(x

i

)+ f

2

p

2

(x

i

)

 

 

1 1

 

 

 

 

 

 

f = n1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

of their weights by the formulas

and

f

2

=

n2

, as well as estimates of the state vectors for

 

1

n

 

 

 

n

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

18

Yu.I. Bogdanov LANL Report quant-ph/0303014

each population ( c(1) and c(2)) according to the algorithm presented above. Then, we will find the component densities

p

 

(x)=

 

c(1)ϕ

(x)

 

2

 

 

 

1

 

 

 

 

 

i

i

 

 

 

(4.5)

p

2

(x)=

 

c(2)ϕ

(x)

 

2

 

 

 

 

 

 

 

i

 

i

 

 

(4.6)

Finally, instead of the initial (prior) estimates of the weights and densities, we will find new (posterior) weights and densities of the mixture components. Applying this (quasiBayesian) procedure repeatedly, we will arrive at a certain equilibrium state when the weights and densities of the components become approximately constant (more precisely, prior and posterior estimates of weights and densities become indistinguishable within statistical fluctuations).

A random-number generator should be used for numerical implementation of the algorithm proposed. Each iteration starts with the setting of the random vector of the length n from a homogeneous distribution on the segment [0,1]. If the same random vector is used at each iteration, the distribution of sample points between components will stop varying after some iterations, i.e., each sample point will correspond to a certain mixture component (perhaps, up to insignificant infinite looping, when a periodic exchange of a few points only happens). In this case, each random vector corresponds to a certain random division of mixture into components that allows modeling the fluctuation in a system.

 

Consider informational aspects of the problem of mixture division into components. The

results presented below are based on the following mathematical inequality:

 

 

f1 p1 ln p1 + f2 p2 ln p2

(f1 p1 + f2 p2 )ln(f1 p1 + f2 p2 ),

(4.7)

that is valid if f1 + f2 =1 , and f1 , f2 , p1 , and p2

are arbitrary nonnegative numbers. We assume

also that 0ln 0 = 0 .

 

 

 

 

 

 

Generally, the following inequality takes place

 

 

s

s

 

s

 

 

 

(fi pi ln pi )(fi

pi )ln

(fi pi ) ,

(4.8)

 

i=1

i=1

 

i=1

 

 

if s

fi =1, and fi , pi

(i =1,..., s) are arbitrary nonnegative numbers.

 

i=1

 

 

 

 

 

 

The equality sign in (4.8) takes place only in two cases: when either the probabilities are equal to each other ( p1 = p2 = ... = ps ) or one of the weights is equal to unity and the other weights are

zero ( fi0 =1, fi = 0, i i0 ). In both cases, the mixed state is reduced to the pure one.

The logarithmic likelihood related to a certain observation xi

in the case when we apply

the structureless approach and the functional L0 is evidently ln(f1 p1(xi )+ f2 p2 (xi )).

In the component approach when we use the functional L1

and randomized (mixed) strategy, the

same observation corresponds to either the logarithmic

likelihood

ln(p1(xi ))

with the

19

Yu.I. Bogdanov

 

LANL Report

 

 

quant-ph/0303014

 

 

 

 

ln(p2 (xi )) with the

probability

 

 

 

 

 

f1 p1 (xi )

 

 

 

 

 

 

or the logarithmic likelihood

f

1

p (x

i

)+ f

2

p

2

(x

i

)

 

 

 

 

 

 

1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

probability

 

 

 

 

f2 p2 (xi

)

 

 

 

 

 

 

. In this case, the mean logarithmic likelihood is given by

f

1

p

(x

i

)+ f

2

p

2

(x

i

)

 

 

 

 

 

1

 

 

 

 

 

 

 

 

 

 

 

 

(xi )ln(p2 (xi ))

 

 

 

 

 

 

 

 

 

 

 

 

 

 

f1 p1

(xi )ln(p1 (xi ))+ f2 p2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

f1 p1 (xi )+ f2 p2

(xi )

 

 

 

 

 

 

 

 

 

 

 

 

 

 

This quantity turns out to be not smaller than the logarithmic likelihood in the first case

 

f1 p1

(xi

)ln(p1 (xi

))+ f2 p2

(xi )ln(p2 (xi ))

ln(f

p

(x

i

)+ f

2

p

2

(x

i

))

(4.9)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

f1 p1 (xi )+ f2 p2 (xi )

1 1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

The validity of the last inequality at arbitrary values of the argument

 

xi

 

follows from the

inequality (4.7). Consider a continuous variable x instead of the discrete one xi

and rewrite the

last inequality in the form

 

f1 p1(x)ln(p1(x))+ f2 p2 (x)ln(p2 (x))

 

(f1 p1 (x)+ f2 p2 (x))ln(f1 p1 (x)+ f2 p2 (x))

(4.10)

 

Integrating with respect to x , we find for the Boltzmann H function (representing the

entropy with the opposite sign [15] )

 

Hmix H0 ,

(4.11)

where

 

H0 = p(x)ln p(x)dx =

 

= (f1 p1 (x)+ f2 p2 (x))ln(f1 p1 (x)+ f2 p2 (x))dx

(4.12)

 

Hmix = f1 p1 (x)ln p1(x)dx + f2 p2 (x)ln p2 (x)dx

(4.13)

Thus, in the component approach, the Boltzmann H function is higher and the entropy

S = −H is lower compared to the structureless description. This means that the representation of data as a mixture of components results in more detailed (and hence, more informative)

description than that in the structureless approach. The difference Imix = Hmix H0 can be

interpreted as information produced in result of dividing the mixture into components. This information is lost in turning from the component description to structureless.

In general case of arbitrary number of mixture components, the mixture density is

s

s

 

p(x)= fi

 

pi (x), where fi =1

(4.14)

i=1

i=1

 

From inequality (4.8), it follows that the component description generally corresponds to

higher (compared to that in the structureless description) value of the Boltzmann

H function

Hmix H0 ,

 

(4.15)

where H0 = p(x)ln p(x)dx

(4.16)

20

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