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In information theory and statistics, Kullback's inequality is a lower bound on the Kullback–Leibler divergence expressed in terms of the large deviations
Kullback's_inequality
Mathematical statistics distance measure
triangle inequality. Numerous references to earlier uses of the symmetrized divergence and to other statistical distances are given in Kullback (1959).
Kullback–Leibler_divergence
Concept in information theory
This fundamental inequality states that the Kullback–Leibler divergence is non-negative. Another inequality concerning the Kullback–Leibler divergence
Inequalities in information theory
Inequalities_in_information_theory
Inequality in information theory
distance) in terms of the Kullback–Leibler divergence. The inequality is tight up to constant factors. Pinsker's inequality states that, if P {\displaystyle
Pinsker's_inequality
Theorem of convex functions
In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral
Jensen's_inequality
Statement in information theory
difference between the two quantities is the Kullback–Leibler divergence or relative entropy, so the inequality can also be written: D K L ( P ‖ Q ) ≡ ∑ i
Gibbs'_inequality
Inequality applying to random variables
In information theory, Fano's inequality (also known as the Fano converse and the Fano lemma) relates the average information lost in a noisy channel to
Fano's_inequality
Lower bound on variance of an estimator
or n + 2 {\displaystyle n+2} . Chapman–Robbins bound Kullback's inequality Brascamp–Lieb inequality Lehmann–Scheffé theorem Ziv–Zakai bound Cramér, Harald
Cramér–Rao_bound
Self-information Kullback–Leibler divergence Entropy estimation Dembo, Amir; Cover, Thomas M.; Thomas, Joy A. (1991). "Information-theoretic inequalities". IEEE
Entropy_power_inequality
Exponentially decreasing bounds on tail distributions of random variables
Markov's inequality or Chebyshev's inequality. The Chernoff bound is related to the Bernstein inequalities. It is also used to prove Hoeffding's inequality, Bennett's
Chernoff_bound
Inequality of sum of product of number and logarithm of ratios
{a}{b}}+k\right)} . The log sum inequality can be used to prove inequalities in information theory. Gibbs' inequality states that the Kullback–Leibler divergence is
Log_sum_inequality
Inequality in information theory
In information theory, the Bretagnolle–Huber inequality bounds the total variation distance between two probability distributions P {\displaystyle P} and
Bretagnolle–Huber_inequality
analysis of variance Kuder–Richardson Formula 20 Kuiper's test Kullback's inequality Kullback–Leibler divergence Kumaraswamy distribution Kurtosis Kushner
List_of_statistics_articles
Lower bound on the log-likelihood of some observed data
{\displaystyle L(\phi ,\theta ;x)} forms a lower bound on the evidence (ELBO inequality) ln p θ ( x ) ≥ E z ∼ q ϕ ( ⋅ | x ) [ ln p θ ( x , z ) q ϕ ( z | x
Evidence_lower_bound
Function that measures dissimilarity between two probability distributions
triangle inequality. For example, the term "Bregman distance" is still found, but "Bregman divergence" is now preferred. Notationally, Kullback & Leibler
Divergence_(statistics)
Metric that measures the distance between two strings of text
(e.g. in contrast to string matching) is fulfillment of the triangle inequality. For example, the strings "Sam" and "Samuel" can be considered to be close
String_metric
Concept in information theory
{KL} }(p^{*}||q)} . This inequality can be interpreted as an information-geometric version of Pythagoras' triangle-inequality theorem, where KL divergence
Information_projection
the concept "quantity of information"" (PDF). Stam, A.J. (1959). "Some inequalities satisfied by the quantities of information of Fisher and Shannon". Information
Quantities_of_information
Similarity of two probability distributions
despite being named a "distance", since it does not obey the triangle inequality. Both the Bhattacharyya distance and the Bhattacharyya coefficient are
Bhattacharyya_distance
Distance between two statistical objects
(symmetry) d(x, z) ≤ d(x, y) + d(y, z) (subadditivity / triangle inequality). Many statistical distances are not metrics, because they lack one or
Statistical_distance
Concept in probability theory
The total variation distance is related to the Kullback–Leibler divergence by Pinsker’s inequality: δ ( P , Q ) ≤ 1 2 D K L ( P ∥ Q ) . {\displaystyle
Total variation distance of probability measures
Total_variation_distance_of_probability_measures
Metric on a smooth statistical manifold
understood to be the infinitesimal form of the relative entropy (i.e., the Kullback–Leibler divergence); specifically, it is the Hessian of the divergence
Fisher_information_metric
Overview of and topical guide to statistics
probability Law of large numbers Central limit theorem Concentration inequality Convergence of random variables Computational statistics Markov chain
Outline_of_statistics
power inequality Etemadi's inequality / (F:R) Gauss's inequality Hoeffding's inequality / (F:R) Khintchine inequality / (F:B) Kolmogorov's inequality / (F:R)
Catalog of articles in probability theory
Catalog_of_articles_in_probability_theory
Probability distribution
ℓ ( α ) {\displaystyle \ell (\alpha )} is strictly concave, by using inequality properties of the polygamma function. Finding the maximum with respect
Gamma_distribution
Concept in information theory
_{i}p_{i}{\left(\ln p_{i}+H(p)\right)}^{2}} . In particular cases inequalities can be proven also by Jensen's inequality: log n = H 0 ≥ H 1 ≥ H 2 ≥ H ∞ . {\displaystyle
Rényi_entropy
Measure of difference between two points
{\displaystyle \Gamma _{n}} that satisfies the data processing inequality must be the Kullback–Leibler divergence. (In fact, a weaker assumption of "sufficiency"
Bregman_divergence
Probability distribution
key role in a multifunctional inequality which implies various bounds for the Dirichlet distribution. Another inequality relates the moment-generating
Dirichlet_distribution
Smooth approximation to the maximum function
(2016). "Guaranteed bounds on the Kullback-Leibler divergence of univariate mixtures using piecewise log-sum-exp inequalities". Entropy. 18 (12): 442. arXiv:1606
LogSumExp
Scientific study of digital information
true metric since it is not symmetric and does not satisfy the triangle inequality (making it a semi-quasimetric). Another interpretation of the KL divergence
Information_theory
Coherent measure for value at risk
and the conditional value at risk (CVaR), obtained from the Chernoff inequality. The EVaR can also be represented by using the concept of relative entropy
Entropic_value_at_risk
Probability distribution
{1}{\lambda }}=\operatorname {\sigma } [X],} in accordance with the median-mean inequality. An exponentially distributed random variable T obeys the relation Pr
Exponential_distribution
Measure of distinguishability between two quantum states
{q_{j}}{p_{j}}}p_{j})=0.} Jensen's inequality also states that equality holds if and only if, for all i, qi = (Σqj) pi, i.e. p = q. Klein's inequality states that the quantum
Quantum_relative_entropy
Relationship of various quantum subsystems
Computation and Quantum Information" "Quantum Entropy and Its Use" Trace Inequalities and Quantum Entropy: An Introductory Course We use the following notation
Strong subadditivity of quantum entropy
Strong_subadditivity_of_quantum_entropy
probability Probability-generating function Vysochanskiï–Petunin inequality Mutual information Kullback–Leibler divergence Le Cam's theorem Large deviations theory
List_of_probability_topics
Measure of dependence between two variables
\operatorname {I} (X;Y)=\operatorname {I} (Y;X)} see below). Using Jensen's inequality on the definition of mutual information we can show that I ( X ; Y )
Mutual_information
Massachusetts – Shannon–Fano coding 1949 – Leon G. Kraft discovers Kraft's inequality, which shows the limits of prefix codes 1949 – Marcel J. E. Golay introduces
Timeline of information theory
Timeline_of_information_theory
Discrete probability distribution
P = Pois ( λ ) {\displaystyle P=\operatorname {Pois} (\lambda )} . Inequalities that relate the cumulative distribution function of a Poisson random
Poisson_distribution
Probability distribution
non-normal random variables uncorrelatedness does not imply independence. The Kullback–Leibler divergence of one normal distribution X 1 ∼ N ( μ 1 , σ 1 2 ) {\textstyle
Normal_distribution
Multi-armed bandit sequential game
from optimal. Multi-armed bandit Design of experiments Concentration inequality Robertson, David S.; Lee, Kevin M.; López-Kolkovska, Beatriz C.; Villar
Best_arm_identification
Notion in statistics
matrix. The Fisher information matrix plays a role in an inequality like the isoperimetric inequality. Of all probability distributions with a given entropy
Fisher_information
Average uncertainty in variable's states
to bound the right side of Shearer's inequality and exponentiate the opposite sides of the resulting inequality you obtain. For integers 0 < k < n let
Entropy_(information_theory)
Concept in information theory
is the Jacobian of the transformation m {\displaystyle m} . The above inequality becomes an equality if the transform is a bijection. Furthermore, when
Differential_entropy
Distance function defined between probability distributions
cost. At this point, a very short proof of the isoperimetric inequality appears. The inequality states that among all open sets of R n {\displaystyle \mathbb
Wasserstein_metric
Separation between two points
the same as the distance from y to x. Distance satisfies the triangle inequality: if x, y, and z are three objects, then d ( x , z ) ≤ d ( x , y ) + d
Distance
Technique in information theory
complex probability expressions into more natural expressions involving the Kullback–Leibler divergence. The topic is considered a traditional method in information
Method_of_types
Information-theoretic measure
distribution p {\displaystyle p} . The definition may be formulated using the Kullback–Leibler divergence D K L ( p ∥ q ) {\displaystyle D_{\mathrm {KL} }(p\parallel
Cross-entropy
Probability distribution
value infinity). The results for higher moments follow from Hölder's inequality, which implies that higher moments (or halves of moments) diverge if lower
Cauchy_distribution
Principle in Bayesian statistics
continuous formulation maximizes relative entropy (equivalently, minimizes Kullback–Leibler divergence) with respect to a specified reference measure or prior
Principle_of_maximum_entropy
Probability distribution
tail of the cumulative distribution function for k ≥ np. Hoeffding's inequality yields the simple bound F ( k ; n , p ) ≤ exp ( − 2 n ( p − k n ) 2
Binomial_distribution
In mathematics, invariant of square matrices
{\sqrt {{\frac {1}{n}}\operatorname {tr} \left(A^{2}\right)}}.} These inequalities can be proved by expressing the traces and the determinant in terms of
Determinant
Branch of probability theory
equipartition property applied to a Bernoulli trial. Then by Chernoff's inequality, it can be shown that P ( M N > x ) < exp ( − N I ( x ) ) {\displaystyle
Large_deviations_theory
Metric used in probability and statistics
{2}}} ). These inequalities follow immediately from the inequalities between the 1-norm and the 2-norm. Statistical distance Kullback–Leibler divergence
Hellinger_distance
Statistic used in signal detection theory
However, d b ′ {\displaystyle d'_{b}} does not satisfy the triangle inequality, so it is not a full metric. In particular, for a yes/no task between
Sensitivity_index
Number taken as representative of a list of numbers
and music. AM, GM, and HM of nonnegative real numbers satisfy these inequalities: A M ≥ G M ≥ H M {\displaystyle \mathrm {AM} \geq \mathrm {GM} \geq \mathrm
Average
Discrete probability distribution
University. Retrieved 2025-01-19. Hoeffding, Wassily (1963). "Probability inequalities for sums of bounded random variables" (PDF). Journal of the American
Hypergeometric_distribution
have dropped the negative sign: the Kullback–Leibler divergence is always non-negative due to Gibbs' inequality. There is an analogy between Shannon's
Information theory and measure theory
Information_theory_and_measure_theory
Statistical NM-method
historical change in social inequality between different educational groups in the US between 1980 and 2010. The trend in inequality was found to be U-shaped
NM-method
Information theory
{\displaystyle I(X;Y|Z)} is the expected (with respect to Z {\displaystyle Z} ) Kullback–Leibler divergence from the conditional joint distribution P ( X , Y )
Conditional mutual information
Conditional_mutual_information
Method of estimating a statistical model's parameters
D_{n+1}}}={\frac {1}{n+1}}\sum _{i=1}^{n+1}\ln {D_{i}}(\theta ).} By the inequality of arithmetic and geometric means, function S n ( θ ) {\displaystyle S_{n}(\theta
Maximum_spacing_estimation
Function that measures dissimilarity between two probability distributions
the measures P and Q coincide. This follows immediately from Jensen’s inequality: D f ( P ∥ Q ) = ∫ f ( d P d Q ) d Q ≥ f ( ∫ d P d Q d Q ) = f ( 1 ) =
F-divergence
Probability distribution
(September 1983). On the similarity of the entropy power inequality and the Brunn Minkowski inequality (PDF). Tech.Report 48, Dept. Statistics, Stanford University
Beta_distribution
Multivalued function in mathematics
Stewart, Seán M. (2009). "On certain inequalities involving the Lambert W function". Journal of Inequalities in Pure & Applied Mathematics. 10 (4) 96
Lambert_W_function
Iterative method for finding maximum likelihood estimates in statistical models
{\theta }}^{(t)}\mid {\boldsymbol {\theta }}^{(t)}).} However, Gibbs' inequality tells us that H ( θ ∣ θ ( t ) ) ≥ H ( θ ( t ) ∣ θ ( t ) ) {\displaystyle
Expectation–maximization algorithm
Expectation–maximization_algorithm
Square matrix containing the distances between elements in a set
= xji), and for any i and j, xij ≤ xik + xkj for all k (the triangle inequality). This can be stated in terms of tropical matrix multiplication When a
Distance_matrix
Application of information theory to thermodynamics and statistical mechanics
Discrimination Information Kullback–Leibler divergence Quantum relative entropy Information theory and measure theory Entropy power inequality Jaynes, E.T. (1957)
Maximum entropy thermodynamics
Maximum_entropy_thermodynamics
Straight path on a curved surface or a Riemannian manifold
generally, a W 1 , 2 {\displaystyle W^{1,2}} curve), the Cauchy–Schwarz inequality gives L ( γ ) 2 ≤ 2 ( b − a ) E ( γ ) {\displaystyle L(\gamma )^{2}\leq
Geodesic
geometric function theory. He introduced Grunsky's theorem and the Grunsky inequalities. Georg Hamel. Oswald Teichmüller German, temporarily employed at OKW
List_of_cryptographers
in KL divergence implies closeness in total variation (via Pinsker's inequality), which in turn implies closeness in Kolmogorov distance. Therefore, a
Distribution_learning_theory
German Signal Intelligence Agency
as part of deciphering of enciphered message, and invented by Solomon Kullback, but simply the first three letters of the word Chiffrierabteilung. From
Cipher Department of the High Command of the Wehrmacht
Cipher_Department_of_the_High_Command_of_the_Wehrmacht
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KULLBACKS INEQUALITY
KULLBACKS INEQUALITY
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KULLBACKS INEQUALITY
KULLBACKS INEQUALITY
KULLBACKS INEQUALITY
KULLBACKS INEQUALITY
KULLBACKS INEQUALITY
KULLBACKS INEQUALITY
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