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Entropy of a process with only two probable values
In information theory, the binary entropy function, denoted H ( p ) {\displaystyle \operatorname {H} (p)} or H b ( p ) {\displaystyle \operatorname
Binary_entropy_function
Model of noisy digital information transfer and storage.
binary symmetric channel (BSC), which has capacity 1 − H b ( P e ) {\displaystyle 1-\operatorname {H} _{\text{b}}(P_{e})} (for the binary entropy function
Binary_erasure_channel
Scientific study of digital information
where Hb is the binary entropy function to the base-2 logarithm: A binary erasure channel (BEC) with erasure probability p is a binary input, ternary output
Information_theory
Average uncertainty in variable's states
In information theory, the entropy of a random variable quantifies the average level of uncertainty or information associated with the variable's potential
Entropy_(information_theory)
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
Common communications channel model
is the binary entropy function. Codes including Forney's code have been designed to transmit information efficiently across the channel. A binary symmetric
Binary_symmetric_channel
Probability distribution modeling a coin toss which need not be fair
sequence of independent Bernoulli trials Bernoulli sampling Binary entropy function Binary decision diagram Uspensky, James Victor (1937). Introduction
Bernoulli_distribution
Function in statistics
the binomial distribution . The logit function is the negative of the derivative of the binary entropy function. The logit is also central to the probabilistic
Logit
Property of a thermodynamic system
Entropy is a thermodynamic state variable that quantifies the probabilistic distribution of accessible microstates in a system. The term and the concept
Entropy
Search algorithm finding the position of a target value within a sorted array
is the binary entropy function and τ {\displaystyle \tau } is the probability that the procedure yields the wrong position. The noisy binary search problem
Binary_search
Measure of algorithmic complexity
{\displaystyle H_{b}} is the binary entropy function (not to be confused with the entropy rate). The Kolmogorov complexity function is equivalent to deciding
Kolmogorov_complexity
Smoothed ramp function
(specifically, the Legendre transformation) of the softplus function is the negative binary entropy function (with base e). This is because (following the definition
Softplus
Lossless data compression scheme
In information theory, an entropy coding (or entropy encoding) is any lossless data compression method that attempts to approach the lower bound declared
Entropy_coding
Family of algorithms for sampling from discrete probability distributions
coding techniques arithmetic we can approach the limit given by the binary entropy function. Donald Knuth, The Art of Computer Programming, Vol 2: Seminumerical
Alias_method
Theory about lossy data compression
where H b {\displaystyle H_{b}} denotes the binary entropy function. Plot of the rate-distortion function for p = 0.5 {\displaystyle p=0.5} : Suppose
Rate–distortion_theory
Number of subsets of a given size
{\displaystyle H(p)=-p\log _{2}(p)-(1-p)\log _{2}(1-p)} is the binary entropy function. It can be further tightened to n 8 k ( n − k ) 2 n H ( k / n )
Binomial_coefficient
Topics referred to by the same term
Nicaragua (IATA:BEF) Belgian franc, a defunct currency (ISO 4217:BEF) Binary entropy function, in information theory Bonus Expeditionary Force, an American veterans'
BEF
Quantum states of two qubits
entropy of entanglement: S r = 1 − h ( p max ) {\displaystyle S_{r}=1-h(p_{\text{max}})} , where h {\displaystyle h} is the binary entropy function.
Bell_diagonal_state
\sum _{i=1}^{k}p_{i}=1} An important special case of this is the binary entropy function: H b ( p ) = H ( p , 1 − p ) = − p log p − ( 1 − p ) log (
Quantities_of_information
Deep learning method
_{\text{ref}}(x)\parallel D(x))} , where H {\displaystyle H} is the binary entropy function, so L ( μ G , μ D ) = − ∫ μ ( d x ) ( H ( ρ ref ( x ) ) + D K L
Generative adversarial network
Generative_adversarial_network
Alloys with high proportions of several metals
High-entropy alloys (HEAs) are alloys that are formed by mixing equal or relatively large proportions of (usually) five or more elements. Prior to the
High-entropy_alloy
other have entropy H ( n m + n ) {\displaystyle H\left({\frac {n}{m+n}}\right)} , where H {\displaystyle H} is the binary entropy function. Here, we use
Graph_entropy
Limit on data transfer rate
{C}{1-H_{2}(p_{b})}}.} and H 2 ( p b ) {\displaystyle H_{2}(p_{b})} is the binary entropy function H 2 ( p b ) = − [ p b log 2 p b + ( 1 − p b ) log 2 ( 1 − p
Noisy-channel_coding_theorem
{\displaystyle {\mathsf {s}}(p)={\frac {{\mathsf {H}}(p)}{1-p}}} for the binary entropy function H ( ⋅ ) {\displaystyle {\mathsf {H}}(\cdot )} . This capacity is
Z-channel (information theory)
Z-channel_(information_theory)
Theory in modern physics that describes gravity as an entropic force
Entropic gravity, also known as emergent gravity, is a theory in modern physics that describes gravity as an entropic force—a force with macro-scale homogeneity
Entropic_gravity
Exponent of a power of two
2 and is the inverse function of the power of two function. There are several alternatives to the log2 notation for the binary logarithm; see the Notation
Binary_logarithm
Operation on mathematical functions
case for a binary (or higher arity) operation. A binary (or higher arity) operation that commutes with itself is called medial or entropic. Composition
Function_composition
Computing concept
95 use CryptoAPI to gather entropy in a manner similar to the Linux kernel's /dev/random. Windows's CryptoAPI uses the binary registry key
Entropy_(computing)
Mathematical functions
(Z_{i})}}}]} The binary cross-entropy compares the observed y ∈ { 0 , 1 } {\displaystyle y\in \{0,1\}} with the predicted probabilities. The average binary cross-entropy
Hyperbolastic_functions
Entropy coding methods
Asymmetric numeral systems (ANS) is a family of entropy encoding methods introduced by Jarosław (Jarek) Duda from Jagiellonian University, used in data
Asymmetric_numeral_systems
Computer programming language
Binary combinatory logic (BCL) is a computer programming language that uses binary terms 0 and 1 to create a complete formulation of combinatory logic
Binary_combinatory_logic
Mathematical function, inverse of an exponential function
however, use this term for an exponential function.) For example, any natural number N can be represented in binary form in no more than log2 N + 1 bits.
Logarithm
In thermodynamics, entropy is a numerical quantity that shows that many physical processes can go in only one direction in time. For example, cream and
Introduction_to_entropy
Method for finding kth smallest value
H(x)=x\log _{2}{\frac {1}{x}}+(1-x)\log _{2}{\frac {1}{1-x}}} is the binary entropy function. The special case of median-finding has a slightly larger lower
Selection_algorithm
While the growth-function is related to the maximum intersection-size, the entropy is related to the average intersection size: Entropy ( H , m ) = E
Growth_function
Family of algorithms used in procedural generation
scanline order. WFC chooses the lowest entropy cell. DV Gen (Apr 17, 2023). Procedural Generation with Wave Function Collapse and Model Synthesis | Unity
Model_synthesis
Exponentially decreasing bounds on tail distributions of random variables
variables. Cramér's theorem Entropic value at risk Hoeffding's inequality Matrix Chernoff bound Moment generating function Boucheron, Stéphane (2013).
Chernoff_bound
Statistical model for a binary dependent variable
where logistic regression is used for binary classification, the MLE minimises the cross-entropy loss function. Logistic regression is an important machine
Logistic_regression
Statistical concept
{\displaystyle 1/H(b)} (here H ( b ) {\displaystyle H(b)} is the binary entropy function). There are algorithms that reach this optimal value in expectation
Fair_coin
Creating sequence of numbers that cannot be predicted
producing the entropy. Usually this process is analog, so a digitizer is used to convert the output of the analog source into a binary representation;
Random_number_generation
Methods of estimating differential entropy given some observations
probability density functions (pdf) is to first make a pdf estimate with some method, and then, from the pdf estimate, compute the entropy. A useful pdf estimate
Entropy_estimation
Type of data structure
binary (two child nodes under each node) but they can just as well use many more child nodes under each node. Usually, a cryptographic hash function such
Merkle_tree
Regression for more than two discrete outcomes
regression, multinomial logit (mlogit), the maximum entropy (MaxEnt) classifier, and the conditional maximum entropy model. Multinomial logistic regression is used
Multinomial logistic regression
Multinomial_logistic_regression
Subject of study in ergodic theory
generators. Thus, for example, the entropy of the Bernoulli process is log 2, since almost every real number has a unique binary expansion. That is, one may
Measure-preserving dynamical system
Measure-preserving_dynamical_system
2.71828...; base of natural logarithms
p(A_{i}).} The function f ( x ) = − x ln x {\displaystyle f(x)=-x\ln x} is thus of fundamental importance, representing the amount of entropy contributed
E_(mathematical_constant)
Smooth approximation of one-hot arg max
function is often used in the final layer of a neural network-based classifier. Such networks are commonly trained under a log loss (or cross-entropy)
Softmax_function
Binary sequence
_{2}{\binom {N}{pN}}\approx NH(p)} where H {\displaystyle H} is the binary entropy function. Thus, the number of bits in this description is 2 ( 1 + ϵ ) log
Algorithmically random sequence
Algorithmically_random_sequence
Unit of information
usually a nibble. In information theory, one bit is the information entropy of a random binary variable that is 0 or 1 with equal probability, or the information
Bit
Increase in the total entropy of a compound system after mixing
In thermodynamics, the entropy of mixing is the increase in the total entropy when several initially separate systems of different composition, each in
Entropy_of_mixing
Establishes the limits to possible data compression
bound on the minimal possible expected length of codewords as a function of the entropy of the input word (which is viewed as a random variable) and of
Shannon's source coding theorem
Shannon's_source_coding_theorem
Computer security technique
{\displaystyle E_{s}} (entropy bits of stack top) E m {\displaystyle E_{m}} (entropy bits of mmap() base) E x {\displaystyle E_{x}} (entropy bits of main executable
Address space layout randomization
Address_space_layout_randomization
Computer data measurements and scales
usages of a few multiplier prefixes in a binary interpretation which has been common in computing until new binary prefixes were defined in the 1990s. The
Orders_of_magnitude_(data)
Dutch computer scientist
Tromp, John (2026-04-26). "The Largest Number Representable in 64 Bits". Entropy. 28 (5): 494. doi:10.3390/e28050494. ISSN 1099-4300. PMC 13205354. PMID 42187911
John_Tromp
Topics referred to by the same term
argument Unary coding, an entropy encoding that represents a number n with n − 1 ones followed by a zero Primary (disambiguation) Binary (disambiguation) This
Unary
entropy loss is limited to n H ( p ) − o ( n ) {\displaystyle nH(p)-o(n)} , where H {\displaystyle H} is the binary entropy function.If min-entropy m
Fuzzy_extractor
Cryptographic device
producing the entropy. Usually this process is analog, so a digitizer is used to convert the output of the analog source into a binary representation;
Hardware random number generator
Hardware_random_number_generator
Physical quantity of hot and cold
point of maximum entropy, the temperature function shows the behavior of a singularity because the slope of the entropy as a function of energy decreases
Temperature
Technique to compress data
occurrence (weight) for each possible value of the source symbol. As in other entropy encoding methods, more common symbols are generally represented using fewer
Huffman_coding
How many different types are in a dataset
applications that use the Shannon entropy. Each log base corresponds to a different measurement unit, which has been called binary digits (bits), decimal digits
Diversity_index
Encoding an n-bit string in m qubits
{\displaystyle m\geq n(1-H(p))} , where H ( p ) {\displaystyle H(p)} is the binary entropy function. Effectively, to recover any bit with high probability, the encoded
Quantum_random_access_code
Probability distribution
differential entropy of X is (measured in nats), the expected value of the negative of the logarithm of the probability density function: h ( X ) = E
Beta_distribution
Power law growth of entropy of language or a stochastic process
is positive and continuous, and the autocorrelation function follows a power law. Shannon entropy – Average uncertainty in variable's statesPages displaying
Hilberg's_hypothesis
Sequential model-based optimization of expensive black-box functions
Ghahramani, Zoubin (2014). "Predictive Entropy Search for Efficient Global Optimization of Black-box Functions". arXiv:1406.2541 [stat.ML]. Hoffman, Matthew
Bayesian_optimization
Set-to-real map with diminishing returns
inequality. Further inequalities for the entropy function are known to hold, see entropic vector. Matroid rank functions Let Ω = { e 1 , e 2 , … , e n } {\displaystyle
Submodular_set_function
Information held in the state of a quantum system
principle, can be stored in a bit of binary strings. Any system having two states is a capable bit. Shannon entropy is the quantification of the information
Quantum_information
Concept in machine learning
p(1\mid x)=0.5} . It's easy to check that the logistic loss and binary cross-entropy loss (Log loss) are in fact the same (up to a multiplicative constant
Loss functions for classification
Loss_functions_for_classification
Model describing the adsorption of a mono-layer of gas molecules on an ideal flat surface
surface dramatically reduces the entropy of the molecular system. To find the entropy decrease, we find the entropy of the molecule when in the adsorbed
Langmuir_adsorption_model
Function that measures dissimilarity between two probability distributions
information geometry, a divergence is a kind of statistical distance: a binary function which establishes the separation from one probability distribution
Divergence_(statistics)
Signal processing computational method
the probability density function of the extracted signal p y {\displaystyle p_{\mathbf {y} }} , then maximizing the joint entropy of Y {\displaystyle Y}
Independent component analysis
Independent_component_analysis
distributions Rice coding: form of entropy coding that is optimal for alphabets following geometric distributions Truncated binary encoding Unary coding: code
List_of_algorithms
Number taken as representative of a list of numbers
interpreted geometrically by using entropy to measure variation: the MLE minimizes cross-entropy (equivalently, relative entropy, Kullback–Leibler divergence)
Average
originally created for and used by Gentoo Linux; Entropy: Used by and created for Sabayon Linux. It works with binary packages that are bzip2-compressed tar archives
List of software package management systems
List_of_software_package_management_systems
Probability distribution in number theory
book}}: |journal= ignored (help) Vepstas, L. (2008), Entropy of Continued Fractions (Gauss-Kuzmin Entropy) (PDF) Weisstein, Eric W. "Gauss–Kuzmin Distribution"
Gauss–Kuzmin_distribution
Mapping arbitrary data to fixed-size values
be mapped to any 3-tuple of hash values. A hash function can be designed to exploit existing entropy in the keys. If the keys have leading or trailing
Hash_function
Mathematical function having a characteristic S-shaped curve or sigmoid curve
step function – Indicator function of positive numbers Logistic regression – Statistical model for a binary dependent variable Logit – Function in statistics
Sigmoid_function
Cryptography number with no hidden properties
could later exploit—though even with natural-seeming selections, enough entropy exists in the possible choices that the utility of these numbers has been
Nothing-up-my-sleeve_number
Compression method for digital images
described mathematically as an iterated function system (IFS). We begin with the representation of a binary image, where the image may be thought of
Fractal_compression
Probability distribution
real-valued random variable. The general form of its probability density function is f ( x ) = 1 2 π σ 2 exp ( − ( x − μ ) 2 2 σ 2 ) . {\displaystyle f(x)={\frac
Normal_distribution
Resistance of a password to being guessed
mathematical ceiling function, i.e. rounding up to the next largest whole number. People are notoriously poor at achieving sufficient entropy to produce satisfactory
Password_strength
Cryptographic hash function
BLAKE is a cryptographic hash function based on Daniel J. Bernstein's ChaCha stream cipher, but a permuted copy of the input block, XORed with round constants
BLAKE_(hash_function)
Index to measure economic inequality
which is the maximum possible entropy of the data minus the observed entropy. It is a special case of the generalized entropy index. It can be viewed as
Theil_index
Feature of systems that defy description
complexity, algorithmic complexity or algorithmic entropy) of a string is the length of the shortest binary program that outputs that string. Minimum message
Complexity
Process of mapping a continuous set to a countable set
approximation can allow the entropy coding design problem to be separated from the design of the quantizer itself. Modern entropy coding techniques such as
Quantization (signal processing)
Quantization_(signal_processing)
Distribution of an uncertain quantity
entropy which, in the case of probability distributions, is the negative expected value of the logarithm of the probability mass or density function or
Prior_probability
Algorithm for creating unique passwords
user's full name is chosen as it provides a sufficiently high level of entropy while being unlikely to be forgotten. master_password: The secret for generating
Master_Password_(algorithm)
Measure for evaluating probabilistic forecasts
Brier/quadaratic score of the sampled cumulative distribution function F {\displaystyle F} for the binary event { X ≤ y } {\displaystyle \{X\leq y\}} . CRPS is
Scoring_rule
Physical law for definition of temperature
provides an independent definition of temperature without reference to entropy, which is defined in the second law. The law was established by Ralph H
Zeroth_law_of_thermodynamics
Decision support tool
formula states the information gain is a function of the entropy of a node of the decision tree minus the entropy of a candidate split at node t of a decision
Decision_tree
Ensemble learning method
aggregating (bagging) Cascading CoBoosting Logistic regression Maximum entropy methods Gradient boosting Margin classifiers Cross-validation List of datasets
Boosting_(machine_learning)
Unreproducible object used in digital security
and 1s in the binary output of a PUF, also referred to as the "Hamming weight" or "bias." A value of 0.5 (50%) indicates maximum entropy, with each output
Physical_unclonable_function
Function that counts distinct factors of a string
logarithm of the complexity function is subadditive. Every real number between 0 and 1 occurs as the topological entropy of some sequence is applicable
Complexity_function
Type of simplified mathematical model
grows. The entropy S {\displaystyle S} associated with this consideration, whether zero or not, is called coarse grained entropy or thermal entropy. A large
Coarse-grained_modeling
Property of functions which is weaker than continuity
most important example is the entropy function, which assigns to each invariant measure its measure-theoretic entropy. Let ( X , T ) {\displaystyle (X
Semi-continuity
Billiard-ball computer Bimetallic strip Bimetric theory Bimoment Binary entropy function Binary pulsar Binary star Binder parameter Binding energy Binet equation Bingham
Index_of_physics_articles_(B)
Lossless data compression method
multiplication and division by 2 can be implemented more efficiently in binary arithmetic. Rice was motivated to propose this simpler subset due to the
Golomb_coding
Universal code which encodes positive integers into binary code words
Fibonacci coding is a universal code which encodes positive integers into binary code words. It is one example of representations of integers based on Fibonacci
Fibonacci_coding
Chart used to show conditions at which physical phases of a substance occur
volume, specific enthalpy, or specific entropy. For example, single-component graphs of temperature vs. specific entropy (T vs. s) for water/steam or for a
Phase_diagram
Type of functions designed for being unsolvable by root-finding algorithms
entropy, and thus just any kind of pseudorandom number generator is insufficient. Ideally, the generation of random numbers in CSPRNGs uses entropy obtained
Cryptographically secure pseudorandom number generator
Cryptographically_secure_pseudorandom_number_generator
Measure of similarity and diversity between sets
Index to positive vectors, where a set corresponds to a binary vector given by the indicator function, i.e. x i ∈ { 0 , 1 } {\displaystyle x_{i}\in \{0,1\}}
Jaccard_index
Type of prefix code
lengths is bounded by a function of the information entropy of the code that, in addition to being bounded, approaches 1 as entropy approaches infinity.
Universal code (data compression)
Universal_code_(data_compression)
BINARY ENTROPY-FUNCTION
BINARY ENTROPY-FUNCTION
BINARY ENTROPY-FUNCTION
BINARY ENTROPY-FUNCTION
BINARY ENTROPY-FUNCTION
BINARY ENTROPY-FUNCTION
BINARY ENTROPY-FUNCTION
BINARY ENTROPY-FUNCTION
BINARY ENTROPY-FUNCTION