Suchergebnisse
Suchergebnisse:
Entropy in information theory is directly analogous to the entropy in statistical thermodynamics. The analogy results when the values of the random variable designate energies of microstates, so Gibbs's formula for the entropy is formally identical to Shannon's formula.
- Prologue
- I(X, Y ) = H(X) + H(Y ) − H(X, Y ),
- I(X, Y ) = H(X) − H(X|Y ) = H(Y ) − H(X|Y ). (2)
- 1.1 Introduction
- 1.2 Probability Spaces and Random Variables
- 1.4 Distributions
- Xi−1(Bi) (1.15)
- PXT (F) = P((XT )−1(F)) = T P(ω : XT (ω) ∈ F); F ∈ BA . (1.17)
- 1.6 Expectation
- 1.7 Asymptotic Mean Stationarity
- 2.1 Introduction
- D(P∥M) ≥ 0
- HP (f) = D(P′∥P × P).
- P(Qj) ln P(Qj) M(Ri)/M(Qj) ,
- X P(ω) P(ω)
- = D(P∥M) − ln 1 = D(P∥M).
- MΦ(ω)
- M(ω)eΦ(ω)
- 2.5 Conditional Entropy and Information
- X = − pX,Y (x, y) ln pX|Y (x|y),
- I(X; Y ) =
- X pX,Y (x, y) ln
- I(X; Y ) = D(PX,Y ∥PX × PY ),
- H(X|Y ) ≥ H(X|Y, Z).
- H(X, f(X, Y )|Y ) =
- I(X; Y |Z) = D(PXY Z∥PX×Y |Z) (2.25)
- X PX×Y |Z(F ×G×D) = P(X ∈ F|Z = z)P(Y ∈ G|Z = z)P(Z = z). (2.26) z∈D
- 1 EM(fn)
- n m(Xn) m(k)(Xn) m(k)(Xn)
- 1 mψ(Xn)
- 4.1 Introduction
- Ri) − P(Qi (Ri − Rj)).
- Qc Q′ K K). (4.2)
- Q′c ⊂
- Theorem 4.2.1 Let {Xn} be a random process with alphabet AX. Let A∞
- 5.2 Divergence
- D(P∥M) = ∞.
- M(Q)
- Z P(Q) ̄ h ≥
- P(F) =
- = EM( |F). dMF dM
- = HPσ(X)∥Mσ(X)(Q).
- F dMF F dM
- Z dPF Z dP/dM
- X |P(Q) − M(Q)| = P(F) − M(F) + M(Fc) − P(Fc) = 2(P(F) − M(F))
- X |P(Q) − M(Q)|
- EM(eΦ) < ∞.
- HP∥M(X, Y ) = HP∥M(Y ) + HP∥M(X|Y ), (5.18)
- Z = ; fX|Y = ,
- XY (F × G) = SY |X(G|x) dPX(x) = PX(F)PY (G);
- MXY Z ≫ PX×Z|Y ≫ PXY Z
- PX×Z|Y (FX × FZ × FY ) = PX|Y (FX|y)dPZY (z, y). (5.23)
- Z Z
- SXY (F) = MX|Y (Fy) dPY (y) = 0
- D(PXY ∥SXY ) = ∞,
- HP∥M(X|Y ) = dPXY ln fX|Y ,
- 1 dPn
- 5.5 Information for General Alphabets
- I(X; Y ) = sup I(q(X); r(Y )),
- I(X; Y ) ≥ 0
- I(X; Y |Z) = ln Z dPXY Z. (5.33) dPX×Y |Z
- Z fXY (x, y)
- PX×Z|Y (FX × FZ × FY ) = d(PX × PY Z). (5.41)
This book is devoted to the theory of probabilistic information measures and their application to coding theorems for information sources and noisy channels. The eventual goal is a general development of Shannon’s mathematical theory of communication, but much of the space is devoted to the tools and methods required to prove the Shannon coding the...
(1) the sum of the two self entropies minus the entropy of the pair. This proved to be the relevant quantity in coding theorems involving more than one distinct random process: the channel coding theorem describing reliable communication through a noisy channel, and the general source coding theorem describing the coding of a source for a user subj...
In this form the mutual information can be interpreted as the information con-tained in one process minus the information contained in the process when the other process is known. While elementary texts on information theory abound with such intuitive descriptions of information measures, we will minimize such discussion because of the potential pi...
An information source or source is a mathematical model for a physical entity that produces a succession of symbols called “outputs” in a random manner. The symbols produced may be real numbers such as voltage measurements from a transducer, binary numbers as in computer data, two dimensional intensity fields as in a sequence of images, continuous ...
measurable space (Ω, B) is a pair consisting of a sample space Ω together with σ-field B of subsets of Ω (also called the event space). A σ-field or σ-algebra is a nonempty collection of subsets of Ω with the following properties:
While in principle all probabilistic quantities associated with a random process can be determined from the underlying probability space, it is often more con-venient to deal with the induced probability measures or distributions on the space of possible outputs of the random process. In particular, this allows us to compare diferent random process...
i∈J i∈J where here we consider Xi as the coordinate functions Xi : AT → A defined by Xi(xT ) = xi. As rectangles in AT are clearly fundamental events, they should be members of any useful σ-field of subsets of AT T . Define the product σ-field BA as the smallest σ-field containing all of the rectangles, that is, the collection of sets that contains...
Such probability measures induced on the outputs of random variables are re-ferred to as distributions for the random variables, exactly as in the simpler case first treated. When T = {m, m + 1, · · · , m + n − 1}, e.g., when we are treating Xn m = (Xn, · · · , Xm+n−1) taking values in An, the distribution is referred to as an n-dimensional or nth ...
Let (Ω, B, m) be a probability space, e.g., the probability space of a directly given random process with alphabet A, (AT , BA , m). T A real-valued random variable f : Ω → R will also be called a measurement since it is often formed by taking a mapping or function of some other set of more general random variables, e.g., the outputs of some random...
A dynamical system (or the associated source) (Ω, B, P, T) is said to be station-ary if
The development of the idea of entropy of random variables and processes by Claude Shannon provided the beginnings of information theory and of the mod-ern age of ergodic theory. We shall see that entropy and related information measures provide useful descriptions of the long term behavior of random pro-cesses and that this behavior is a key facto...
(2.6) with equality if and only if P = M. In this form the result is known as the divergence inequality. The fact that the divergence of one probability measure with respect to another is nonnegative and zero only when the two measures are the same suggest the interpretation of divergence as a “distance” between the two probability measures, that i...
Note that if we let X and Y be the coordinate random variables on our product space, then both P′ and P × P give the same marginal probabilities to X and Y , that is, PX = PY = P. P′ is an extreme distribution on (X, Y ) in the sense that with probability one X = Y ; the two coordinates are deterministically dependent on one another. P × P, however...
i:Ri⊂Qj where we also used the fact that M(Qj) cannot be 0 since then P(Qj) would also have to be zero. Since Ri ⊂ Qj, P(Ri)/P(Qj) = P(Ri T Qj)/P(Qj) = P(Ri|Qj) is an elementary conditional probability. Applying a similar argument to M and dividing by P(Qj), the above expression becomes
EP Φ − ln(EM(eΦ)) = P(ω) ln − ln(X M(ω) ) M(ω) M(ω) ω ω
This proves that the supremum over all Φ is no smaller than the divergence. To prove the other half observe that for any bounded random variable Φ, EP Φ − ln EM(eΦ) = EP eΦ ln EM(eΦ)
ln , M(ω) where the probability measure MΦ is defined by
MΦ(ω) = . P x M(x)eΦ(x) We now have for any Φ that D(P∥Q) − EP Φ − ln(EM(eΦ)) = X P(ω)
We now turn to other notions of information. While we could do without these if we confined interest to finite alphabet processes, they will be essential for later generalizations and provide additional intuition and results even in the finite alphabet case. We begin by adding a second finite alphabet measurement to the setup of the previous sectio...
x,y where pX,Y (x, y) is the joint pmf for (X, Y ) and pX|Y (x|y) = pX,Y (x, y)/pY (y) is the conditional pmf. Defining
= H(X) + H(Y ) − H(X, Y ) H(X) − H(X|Y ) = H(Y ) − H(Y |X). In terms of distributions and pmf’s we have that I(X; Y ) =
(y) x,y pX(x)pY x,y pX(x) = X pY |X(y|x) pX,Y (x, y) ln . pY (y) x,y Note also that mutual information can be expressed as a divergence by
where PX × PY is the product measure on X, Y , that is, a probability measure which gives X and Y the same marginal distributions as PXY , but under which X and Y are independent. Entropy is a special case of mutual information since
Comments: The first relation has the interpretation that given a random vari-able, there is no additional information in a measurement made on the random variable. The second and third relationships follow from the first and the def-initions. The third relation is a form of chain rule and it implies that given a measurement on a random variable, th...
= H(X, f(X, Y ), Y ) − H(Y ) H(X, Y ) − H(Y ) = H(X|Y ). The final relation follows from the second by replacing Y by Y, Z and setting g(Y, Z) = Y . 2 In a similar fashion we can consider conditional relative entropies. Suppose now that M and P are two probability measures on a common space, that X and Y are two random variables defined on that spa...
where PX×Y |Z is the distribution defined by its values on rectangles as
FX×FZ×FY d(PX × PY ) The thrust of the proof is the demonstration that for any measurable nonnega-tive function f(x, z)
FX×FZ×FY d(PX × PY ) The thrust of the proof is the demonstration that for any measurable nonnega-tive function f(x, z)
FX×FZ×FY d(PX × PY ) The thrust of the proof is the demonstration that for any measurable nonnega-tive function f(x, z)
FX×FZ×FY d(PX × PY ) The thrust of the proof is the demonstration that for any measurable nonnega-tive function f(x, z)
FX×FZ×FY d(PX × PY ) The thrust of the proof is the demonstration that for any measurable nonnega-tive function f(x, z)
FX×FZ×FY d(PX × PY ) The thrust of the proof is the demonstration that for any measurable nonnega-tive function f(x, z)
FX×FZ×FY d(PX × PY ) The thrust of the proof is the demonstration that for any measurable nonnega-tive function f(x, z)
FX×FZ×FY d(PX × PY ) The thrust of the proof is the demonstration that for any measurable nonnega-tive function f(x, z)
FX×FZ×FY d(PX × PY ) The thrust of the proof is the demonstration that for any measurable nonnega-tive function f(x, z)
FX×FZ×FY d(PX × PY ) The thrust of the proof is the demonstration that for any measurable nonnega-tive function f(x, z)
FX×FZ×FY d(PX × PY ) The thrust of the proof is the demonstration that for any measurable nonnega-tive function f(x, z)
FX×FZ×FY d(PX × PY ) The thrust of the proof is the demonstration that for any measurable nonnega-tive function f(x, z)
FX×FZ×FY d(PX × PY ) The thrust of the proof is the demonstration that for any measurable nonnega-tive function f(x, z)
FX×FZ×FY d(PX × PY ) The thrust of the proof is the demonstration that for any measurable nonnega-tive function f(x, z)
FX×FZ×FY d(PX × PY ) The thrust of the proof is the demonstration that for any measurable nonnega-tive function f(x, z)
FX×FZ×FY d(PX × PY ) The thrust of the proof is the demonstration that for any measurable nonnega-tive function f(x, z)
FX×FZ×FY d(PX × PY ) The thrust of the proof is the demonstration that for any measurable nonnega-tive function f(x, z)
FX×FZ×FY d(PX × PY ) The thrust of the proof is the demonstration that for any measurable nonnega-tive function f(x, z)
FX×FZ×FY d(PX × PY ) The thrust of the proof is the demonstration that for any measurable nonnega-tive function f(x, z)
FX×FZ×FY d(PX × PY ) The thrust of the proof is the demonstration that for any measurable nonnega-tive function f(x, z)
FX×FZ×FY d(PX × PY ) The thrust of the proof is the demonstration that for any measurable nonnega-tive function f(x, z)
FX×FZ×FY d(PX × PY ) The thrust of the proof is the demonstration that for any measurable nonnega-tive function f(x, z)
FX×FZ×FY d(PX × PY ) The thrust of the proof is the demonstration that for any measurable nonnega-tive function f(x, z)
FX×FZ×FY d(PX × PY ) The thrust of the proof is the demonstration that for any measurable nonnega-tive function f(x, z)
FX×FZ×FY d(PX × PY ) The thrust of the proof is the demonstration that for any measurable nonnega-tive function f(x, z)
FX×FZ×FY d(PX × PY ) The thrust of the proof is the demonstration that for any measurable nonnega-tive function f(x, z)
FX×FZ×FY d(PX × PY ) The thrust of the proof is the demonstration that for any measurable nonnega-tive function f(x, z)
FX×FZ×FY d(PX × PY ) The thrust of the proof is the demonstration that for any measurable nonnega-tive function f(x, z)
FX×FZ×FY d(PX × PY ) The thrust of the proof is the demonstration that for any measurable nonnega-tive function f(x, z)
FX×FZ×FY d(PX × PY ) The thrust of the proof is the demonstration that for any measurable nonnega-tive function f(x, z)
FX×FZ×FY d(PX × PY ) The thrust of the proof is the demonstration that for any measurable nonnega-tive function f(x, z)
FX×FZ×FY d(PX × PY ) The thrust of the proof is the demonstration that for any measurable nonnega-tive function f(x, z)
FX×FZ×FY d(PX × PY ) The thrust of the proof is the demonstration that for any measurable nonnega-tive function f(x, z)
FX×FZ×FY d(PX × PY ) The thrust of the proof is the demonstration that for any measurable nonnega-tive function f(x, z)
FX×FZ×FY d(PX × PY ) The thrust of the proof is the demonstration that for any measurable nonnega-tive function f(x, z)
FX×FZ×FY d(PX × PY ) The thrust of the proof is the demonstration that for any measurable nonnega-tive function f(x, z)
FX×FZ×FY d(PX × PY ) The thrust of the proof is the demonstration that for any measurable nonnega-tive function f(x, z)
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The statistical definition of entropy defines it in terms of the statistics of the motions of the microscopic constituents of a system – modeled at first classically, e.g. Newtonian particles constituting a gas, and later quantum-mechanically (photons, phonons, spins, etc.). The two approaches form a consistent, unified view of the same ...
19. Okt. 2021 · This Special Issue, “The Statistical Foundations of Entropy”, is dedicated to discussing solutions and delving into concepts, methods, and algorithms for improving our understanding of the statistical foundations of entropy in complex systems, with a particular focus on the so-called generalized entropies that go beyond the usual Boltzmann–Gibbs...
- Petr Jizba, Jan Korbel
- 2021
Entropy is a thermodynamic property just like pressure, volume, or temperature. Therefore, it connects the microscopic and the macroscopic world view. Boltzmann's principle is regarded as the foundation of statistical mechanics. Gibbs entropy formula. The macroscopic state of a system is characterized by a distribution on the microstates.
Entropy and Information Theory. It was shown in the classic 1948 work of Claude Shannon that entropy is in fact a measure of information5.
19. Okt. 2021 · During the last few decades, the notion of entropy has become omnipresent in many scientific disciplines, ranging from traditional applications in statistical physics and chemistry, information theory, and statistical estimation to more recent applications in biology, astrophysics, geology, financial markets, or social networks.
