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Concept in information theory
Differential entropy (also referred to as continuous entropy) in information theory is a property of absolutely continuous probability distributions which
Differential_entropy
Average uncertainty in variable's states
that entropy should be a measure of how informative the average outcome of a variable is. For a continuous random variable, differential entropy is analogous
Entropy_(information_theory)
Measure of distance to normality
(unlike differential entropy, which can be negative). There is a physical quantity closely linked to free energy (free enthalpy), with a unit of entropy and
Negentropy
Measure of information in probability and information theory
in which case we say that the differential entropy is not defined. As in the discrete case the joint differential entropy of a set of random variables
Joint_entropy
Measure of relative information in probability theory
In information theory, the conditional entropy quantifies the amount of information needed to describe the outcome of a random variable Y {\displaystyle
Conditional_entropy
Term in information theory
d-dimensional entropy does not necessarily exist there. Finally, dimensional-rate bias generalizes the Shannon's entropy and differential entropy, as one could
Information_dimension
properties; for example, differential entropy may be negative. The differential analogies of entropy, joint entropy, conditional entropy, and mutual information
Quantities_of_information
Principle in Bayesian statistics
(or an explicitly specified prior weighting). In continuous cases, differential entropy depends on the choice of coordinates and is not invariant under reparameterization
Principle_of_maximum_entropy
Information-theoretic measure
In information theory, the cross-entropy between two probability distributions p {\displaystyle p} and q {\displaystyle q} , over the same underlying
Cross-entropy
Probability distribution
the discrete entropy. It is known since then that the differential entropy may differ from the infinitesimal limit of the discrete entropy by an infinite
Beta_distribution
Methods of estimating differential entropy given some observations
learning, and time delay estimation it is useful to estimate the differential entropy of a system or process, given some observations. The simplest and
Entropy_estimation
Probability distribution that has the most entropy of a class
with probability density p ( x ) {\displaystyle p(x)} , then the differential entropy of X {\displaystyle X} is defined as H ( X ) = − ∫ − ∞ ∞ p ( x )
Maximum entropy probability distribution
Maximum_entropy_probability_distribution
Probability distribution
{\displaystyle \operatorname {erf} (z)} is the error function. The differential entropy is given by[citation needed] H = 1 + ln ( σ 2 ) + γ 2 {\displaystyle
Rayleigh_distribution
Mathematical statistics distance measure
distributions, for which Shannon entropy ceases to be so useful (see differential entropy), but the relative entropy continues to be just as relevant
Kullback–Leibler_divergence
Time density of the average information in a stochastic process
In the mathematical theory of probability, the entropy rate or source information rate of a stochastic process is, informally, the time density of the
Entropy_rate
Notion in information theory
Shannon for differential entropy. It was formulated by Edwin Thompson Jaynes to address defects in the initial definition of differential entropy. Shannon
Limiting density of discrete points
Limiting_density_of_discrete_points
Scientific study of digital information
quantities include the Rényi entropy and the Tsallis entropy (generalizations of the concept of entropy), differential entropy (a generalization of quantities
Information_theory
Establishes the limits to possible data compression
its time series X1, ..., Xn is i.i.d. with entropy H(X) in the discrete-valued case and differential entropy in the continuous-valued case. The Source
Shannon's source coding theorem
Shannon's_source_coding_theorem
Theory about lossy data compression
with finite differential entropy, R ( D ) ≥ h ( X ) − h ( D ) {\displaystyle R(D)\geq h(X)-h(D)\,} where h(D) is the differential entropy of a Gaussian
Rate–distortion_theory
Topics referred to by the same term
Entropy (information theory), also called Shannon entropy, a measure of the unpredictability or information content of a message source Differential entropy
Entropy_(disambiguation)
Generalization of the one-dimensional normal distribution to higher dimensions
vector, it is distributed as a generalized chi-squared variable. The differential entropy of the multivariate normal distribution is h ( f ) = − ∫ − ∞ ∞ ∫
Multivariate normal distribution
Multivariate_normal_distribution
closely related with other notions of entropy found in dynamical systems and plays an important role in differential geometry and geometric group theory
Volume_entropy
Measure of dependence between two variables
variable. The concept of mutual information is intimately linked to that of entropy of a random variable, a fundamental notion in information theory that quantifies
Mutual_information
Use of the second law of thermodynamics to distinguish past from future
Entropy is one of the few quantities in the physical sciences that requires a particular direction for time, sometimes called an arrow of time. As one
Entropy_as_an_arrow_of_time
Property of a thermodynamic system
Clausius named the concept of "the differential of a quantity which depends on the configuration of the system" entropy (Entropie) after the Greek word for
Entropy
Probability distribution
the new distribution. The differential entropy of the half-normal distribution is exactly one bit less the differential entropy of a zero-mean normal distribution
Half-normal_distribution
Measure of particle positions within a system
V, Gilson MK (March 2010). "Thermodynamic and Differential Entropy under a Change of Variables". Entropy. 12 (3): 578–590. Bibcode:2010Entrp..12..578H
Configuration_entropy
Probability distribution
distributions on R {\textstyle \mathbb {R} } with finite differential entropy, and H(P) is the differential entropy of P. This follows immediately from the observation
Student's_t-distribution
Equation in statistical mechanics
Boltzmann equation, which is a partial differential equation) is a probability equation relating the entropy S {\displaystyle S} , also written as S
Boltzmann's_entropy_formula
Topic in mathematics
{\displaystyle H} is simply the entropy of a symbol) and the continuous-valued case (where H {\displaystyle H} is the differential entropy instead). The definition
Asymptotic equipartition property
Asymptotic_equipartition_property
Basic noise model used in information theory
I ( X ; Y ) {\displaystyle I(X;Y)} , writing it in terms of the differential entropy: I ( X ; Y ) = h ( Y ) − h ( Y ∣ X ) = h ( Y ) − h ( X + Z ∣ X )
Additive_white_Gaussian_noise
Theorem that tells the maximum rate at which information can be transmitted
Information theory Entropy Differential entropy Conditional entropy Joint entropy Mutual information Directed information Conditional mutual information
Shannon–Hartley_theorem
Branch of mathematics
combinatorics Continuous probability Differential entropy in information theory Differential games Differential geometry, the application of calculus
Mathematical_analysis
Probability distribution and special case of gamma distribution
being σ 2 = 2 k n {\textstyle \sigma ^{2}={\tfrac {2k}{n}}} ). The differential entropy is given by h = ∫ 0 ∞ f ( x ; k ) ln f ( x ; k ) d x = k 2 + ln
Chi-squared_distribution
Application of information theory to thermodynamics and statistical mechanics
of entropy for microscopic statistical mechanical accounts is also lacking. Technical note: For the reasons discussed in the article differential entropy
Maximum entropy thermodynamics
Maximum_entropy_thermodynamics
Classic entropy of a quantum-mechanical density matrix
Neumann entropy, but unlike the classical differential entropy which can be negative at low temperature. In fact, the minimum value of the Wehrl entropy is
Wehrl_entropy
Probability distribution
\operatorname {Dir} ({\boldsymbol {\alpha }})} random variable, the differential entropy of X (in nat units) is h ( X ) = E [ − ln f ( X ) ] = ln B
Dirichlet_distribution
Probability distribution
distribution with λ = 1/μ has the largest differential entropy. In other words, it is the maximum entropy probability distribution for a random variate
Exponential_distribution
Quantity in information theory
itself. For continuous random variables the corresponding concept is differential entropy. This measure has also been called surprisal, as it represents the
Information_content
Probability distribution
\,\sec ^{2}\left[\pi \left(p-{\tfrac {1}{2}}\right)\right].} The differential entropy of a distribution can be defined in terms of its quantile density
Cauchy_distribution
American academic (1922–1998)
students included Joseph H. Eberly and Douglas James Scalapino. Differential entropy Limiting density of discrete points Clark J.W.; Norberg R.E.; Bretthorst
Edwin_Thompson_Jaynes
Physical law for entropy and heat
entropy. Uhlenbeck, G. E.; Ford, G. W. (1963), p. 16. Carathéodory, C. (1909). Buchdahl, H.A. (1966), p. 68. Sychev, V. V. (1991). The Differential Equations
Second_law_of_thermodynamics
Signal processing computational method
find the most nongaussian variables. A simple proof can be found in Differential entropy. J ( x ) = S ( y ) − S ( x ) {\displaystyle J(x)=S(y)-S(x)\,} y is
Independent component analysis
Independent_component_analysis
Monte Carlo algorithm
posterior mutual information, posterior differential entropy, and posterior conditional differential entropy, respectively. We can similarly define information
Gibbs_sampling
Influence of thermodynamics on evolution
S=\int {dQ \over \tau }} where S = {\displaystyle S=} entropy d Q = {\displaystyle dQ=} a differential amount of heat passed into a thermodynamic system τ
Entropy_and_life
R n → R {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} } , the differential entropy of X {\displaystyle X} , denoted h ( X ) {\displaystyle h(X)} , is
Entropy_power_inequality
Thermodynamic potential of entropy, analogous to the free energy
A thermodynamic free entropy is an entropic thermodynamic potential analogous to the free energy. Also known as a Massieu, Planck, or Massieu–Planck potentials
Free_entropy
Information-theoretical limit on transmission rate in a communication channel
Y_{2}|X_{1},X_{2})\end{aligned}}} Let us rewrite the last term of entropy. H ( Y 1 , Y 2 | X 1 , X 2 ) = ∑ ( x 1 , x 2 ) ∈ X 1 × X 2 P ( X 1 , X
Channel_capacity
Interpretation of entropy as the change in arrangement of a system's particles
alterations is quantified energetically by a measure of "entropy" change, according to the following differential expression: ∫ δ Q T ≥ 0 {\displaystyle \int \!{\frac
Entropy_(order_and_disorder)
Type of set in information theory
H ( X ) {\displaystyle H(X)} being the entropy rate. If the process is continuous valued, differential entropy is used instead. Counter-intuitively, the
Typical_set
Probability distribution on a hyper-sphere of arbitrary dimension
The expected value can be used to compute differential entropy and KL divergence. The differential entropy of VMF ( μ , κ ) {\displaystyle {\text{VMF}}({\boldsymbol
Von_Mises–Fisher_distribution
Information theory
necessary in the symbol for joint entropy, since the joint entropy of any number of random variables is the same as the entropy of their joint distribution
Conditional mutual information
Conditional_mutual_information
Technique for solving differential equations
integrating factor that makes entropy an exact differential. An integrating factor is any expression that a differential equation is multiplied by to facilitate
Integrating_factor
Family of continuous probability distributions
Victor; Nichols, Jonathan M.; Bucholtz, Frank (2013). Handbook of Differential Entropy. Chapman and Hall/CRC. p. 100. ISBN 978-1-4665-8317-7. Jones, M.C
Kumaraswamy_distribution
Unit of information
possibilities for a real number chosen between 0 and 1, so-called differential entropy can be used to quantify the information content of an analog signal
Shannon_(unit)
Observational basis of thermodynamics
define a group of physical quantities, such as temperature, energy, and entropy, that characterize thermodynamic systems in thermodynamic equilibrium.
Laws_of_thermodynamics
System where changes of output are not proportional to changes of input
equations in which the unknowns (or the unknown functions in the case of differential equations) appear as variables of a polynomial of degree higher than
Nonlinear_system
Signal encoder
of 2 to 4 can be achieved if differences are subsequently entropy coded, because the entropy of the difference signal is much smaller than that of the
Differential pulse-code modulation
Differential_pulse-code_modulation
Distributed source coding concept
than their joint entropy H ( X , Y ) {\displaystyle H(X,Y)} and none of the sources is encoded with a rate smaller than its entropy, distributed coding
Slepian–Wolf_coding
Irreversible transformation of energy into forms less capable of doing work
an associated increase in entropy. Processes with defined local temperature produce entropy at a certain rate. The entropy production rate times local
Dissipation
whereas for continuous random variables the related concept of differential entropy, written h ( X ) {\displaystyle h(X)} , is used (see Cover and Thomas
Information theory and measure theory
Information_theory_and_measure_theory
Thermodynamic theorem
nearly-ideal gas of molecules. As this quantity H was meant to represent the entropy of thermodynamics, the H-theorem was an early demonstration of the power
H-theorem
Concept in information theory
on the sum of the (differential) Rényi entropies, Hα(|f|2)+Hβ(|g|2) , where 1/α + 1/β = 2, which generalize the Shannon entropies. For simplicity, we
Entropic_uncertainty
engineering, physics and mathematics. Many are integral operators and differential operators. In the following L is an operator L : F → G {\displaystyle
List_of_mathematic_operators
Technical note: For the reasons formed in the article differential entropy, the simple format of Shannon entropy ceases to be directly formatted for random variables
Minimum energy performance standard
Minimum_energy_performance_standard
Continuous probability distribution
asymmetric Laplace distribution with parameters (m1-m2, λ, κ) The differential entropy of the ALD is H = − ∫ − ∞ ∞ f A L ( x ) log ( f A L ( x ) ) d x
Asymmetric Laplace distribution
Asymmetric_Laplace_distribution
Mathematical model which approximates the behavior of real gases
the entropy is an exact differential, using the chain rule, the change in entropy when going from a reference state 0 to some other state with entropy S
Ideal_gas
Austrian mathematician and theoretical physicist (1844–1906)
law of thermodynamics. In 1877, he provided the current definition of entropy, S = k B ln Ω {\displaystyle S=k_{\rm {B}}\ln \Omega } , where Ω is the
Ludwig_Boltzmann
Cryptographic device
generates random numbers from a physical process capable of producing entropy, unlike a pseudorandom number generator (PRNG) that utilizes a deterministic
Hardware random number generator
Hardware_random_number_generator
Type of energy transfer
temperature T form the exact differential d S = δ Q T , {\displaystyle \mathrm {d} S={\frac {\delta Q}{T}},} and that S, the entropy of the working body, is
Heat
State-dependent measures that converge to the mutual information
s i {\displaystyle \mathrm {I_{si}} } , is defined by a difference of entropies, I s i ( X ; Y = y ) ≡ H ( X ) − H ( X | Y = y ) {\displaystyle \mathrm
State-dependent_information
Development of entropy in a thermodynamic system
Entropy production (or generation) is the amount of entropy which is produced during heat process to evaluate the efficiency of the process. Entropy is
Entropy_production
Specific mathematical differential form
An inexact differential or imperfect differential is a differential whose integral is path dependent. It is most often used in thermodynamics to express
Inexact_differential
Class of confidence intervals around statistical functionals of a distribution
Learned-Miller, E.; DeStefano, J. (2008). "A probabilistic upper bound on differential entropy". IEEE Transactions on Information Theory. 54 (11): 5223–5230. arXiv:cs/0504091
CDF-based nonparametric confidence interval
CDF-based_nonparametric_confidence_interval
Diagnostic odds ratio Dickey–Fuller test Difference in differences Differential entropy Diffusion process Diffusion-limited aggregation Digby's H Dimension
List_of_statistics_articles
Problem in information theory and communication
than their joint entropy H ( X , Y ) {\displaystyle H(X,Y)} and none of the sources is encoded with a rate larger than its entropy, distributed coding
Distributed_source_coding
Concept of vector calculus
and differential topology, a closed form is a differential form α whose exterior derivative is zero (dα = 0); and an exact form is a differential form
Closed and exact differential forms
Closed_and_exact_differential_forms
Energy dissipation and entropy production extremal principles are ideas developed within non-equilibrium thermodynamics that attempt to predict the likely
Extremal principles in non-equilibrium thermodynamics
Extremal_principles_in_non-equilibrium_thermodynamics
Branch of thermodynamics
time rate of entropy production. Theoretical analysis shows that chemical reactions do not obey extremal principles for the second differential of time rate
Non-equilibrium thermodynamics
Non-equilibrium_thermodynamics
Maxwell–Boltzmann distribution for an ideal gas, and the implications of the Entropy quantity. The distribution is valid for atoms or molecules constituting
Table of thermodynamic equations
Table_of_thermodynamic_equations
Thermodynamic cycle
and a longer expansion stroke. The first Atkinson-cycle engine, the differential engine, used opposed pistons. The second and best-known design was the
Atkinson_cycle
Physical quantity of hot and cold
including the macroscopic entropy, though microscopically referable to the Gibbs statistical mechanical definition of entropy for the canonical ensemble
Temperature
Branch of physics which studies the behavior of materials modeled as continuous media
{\rho ~s}{T}}~{\text{dV}}.}} We can show that the entropy inequality may be written in differential form as ρ η ˙ ≥ − ∇ ⋅ ( q T ) + ρ s T . {\displaystyle
Continuum_mechanics
Energy contained within a system
the founders of thermodynamics who is also the author of the notion of entropy as a state variable. The total internal energy of a system cannot practically
Internal_energy
On finding a maximal set of solutions of a system of first-order homogeneous linear PDEs
of an overdetermined system of first-order homogeneous linear partial differential equations. In modern geometric terms, given a family of vector fields
Frobenius theorem (differential topology)
Frobenius_theorem_(differential_topology)
Relation between thermodynamic states
existence of entropy as a state function S {\displaystyle S} whose differential d S {\displaystyle dS} is proportional to the heat differential form δ Q {\displaystyle
Adiabatic_accessibility
Function describing equilibrium states of a system
system into a different equilibrium state. Internal energy, enthalpy, and entropy are examples of state quantities or state functions because they quantitatively
State_function
American mathematician and Nobel Laureate (1928–2015)
contributions to game theory, real algebraic geometry, differential geometry, and partial differential equations. Nash and fellow game theorists John Harsanyi
John_Forbes_Nash_Jr.
Thermodynamic quantity
will yield the exact differential of an entropy state function dS = δQ/T. Thermodynamics Sychev, V. V. (1991). The Differential Equations of Thermodynamics
Process_function
Relation between temperature and the equilibrium constant of a chemical reaction
equation, is especially effective in estimating the change in enthalpy and entropy of a chemical reaction. The standard pressure, P 0 {\displaystyle P^{0}}
Van_'t_Hoff_equation
Type of thermodynamic potential
{\displaystyle H} is the enthalpy of the system S {\displaystyle S} is the entropy of the system T {\displaystyle T} is the temperature of the system V {\displaystyle
Gibbs_free_energy
Partial differential relations in thermodynamics
(absolute temperature), V {\displaystyle V} (volume), and S {\displaystyle S} (entropy). They are named for the physicist James Clerk Maxwell, who first presented
Maxwell_relations
Device for measuring a physical quantity
or coffee cup calorimeter Differential Scanning Calorimeter Reaction calorimeter See also Calorimeter or Calorimetry Entropy is accessible indirectly by
List_of_measuring_instruments
2011 book by Terrence Deacon
laser light. Maximum entropy production: The organized structure of a morphodynamic system forms to facilitate maximal entropy production. In the case
Incomplete_Nature
Model of fluid flow through a frictionless constant-area duct with heat transfer
calorically perfect flows the maximum entropy occurs at M = 1. The Rayleigh flow model begins with a differential equation that relates the change in Mach
Rayleigh_flow
Relations between flows and forces, or gradients, in thermodynamic systems
probability distribution function can be expressed through the second differential of the entropy w = A ~ e − 1 2 β i k x i x k ; β i k = β k i = − 1 k ∂ 2 S ∂
Onsager_reciprocal_relations
Concept in information theory
g(y)=\int _{-\infty }^{\infty }f(x)e^{-2\pi ixy}\,dx,} the sum of the differential entropies of | f | 2 {\displaystyle |f|^{2}} and | g | 2 {\displaystyle |g|^{2}}
Inequalities in information theory
Inequalities_in_information_theory
Initial estimate or framework to the solution of a mathematical problem
thought of as a "trial answer" and an important technique in solving differential equations, possibly with initial or boundary conditions. After an ansatz
Ansatz
Thought experiment in statistical physics
semi-classical derivation of entropy that does not take into account the indistinguishability of particles yields an expression for entropy which is not extensive
Gibbs_paradox
Reiteration of the second law of thermodynamics
from the properties of an exact differential (see equation 8 in the exact differential article) and from the energy/entropy equation of state that, for a
Principle_of_minimum_energy
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DIFFERENTIAL ENTROPY
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