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Vector with non-negative entries that add up to one
statistics, a probability vector or stochastic vector is a vector with non-negative entries that add up to one. Underlying every probability vector is an experiment
Probability_vector
Matrix used to describe the transitions of a Markov chain
{\displaystyle \leq 1.} In the same vein, one may define a probability vector as a vector whose elements are nonnegative real numbers which sum to 1.
Stochastic_matrix
Random process independent of past history
become note or pitch values, and a probability vector for each note is constructed, completing a transition probability matrix (see below). An algorithm
Markov_chain
Branch of mathematics concerning probability
Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations
Probability_theory
Inference algorithm for hidden Markov models
occurring. Given an arbitrary row-vector describing the state of the system ( π {\displaystyle \mathbf {\pi } } ), the probability of observing event j is then:
Forward–backward_algorithm
Mathematical function for the probability a given outcome occurs in an experiment
In probability theory and statistics, a probability distribution describes how probabilities are assigned to the possible results of a random phenomenon—more
Probability_distribution
Broad concept generalizing scalars in mathematics and physics
In mathematics and physics, a vector is a generalization of a single number. It may denote a vector quantity, i.e., physical quantity that cannot be expressed
Vector (mathematics and physics)
Vector_(mathematics_and_physics)
Description of continuous random distribution
In probability theory, a probability density function (PDF), density function, or simply density of an absolutely continuous random variable, is a function
Probability_density_function
Complex number whose squared absolute value is a probability
form of S-matrices. Whereas moduli of vector components squared, for a given vector, give a fixed probability distribution, moduli of matrix elements
Probability_amplitude
Distributions in probability theory
It is a compound probability distribution, where a probability vector p is drawn from a Dirichlet distribution with parameter vector α {\displaystyle
Dirichlet-multinomial distribution
Dirichlet-multinomial_distribution
Value for the flow of probability in quantum mechanics
one thinks of probability as a heterogeneous fluid, then the probability current is the rate of flow of this fluid. It is a real vector that changes with
Probability_current
Concept in probability theory
In Bayesian probability theory, if, given a likelihood function p ( x ∣ θ ) {\displaystyle p(x\mid \theta )} , the posterior distribution p ( θ ∣ x )
Conjugate_prior
Random variable with multiple component dimensions
In probability and statistics, a multivariate random variable or random vector is a list or vector of mathematical variables each of whose value is unknown
Multivariate_random_variable
Measure of similarity and diversity between sets
which is called the "Probability" Jaccard. It has the following bounds against the Weighted Jaccard on probability vectors. J W ( x , y ) ≤ J P ( x
Jaccard_index
Generalization of the one-dimensional normal distribution to higher dimensions
k=\operatorname {rank} \left(\Sigma \right)=2} ), the probability density function of a vector [XY] ′ {\displaystyle {\text{[XY]}}\prime } is: f ( x
Multivariate normal distribution
Multivariate_normal_distribution
of genetic algorithm where the genotype of an entire population (probability vector) is evolved rather than individual members. The algorithm is proposed
Population-based incremental learning
Population-based_incremental_learning
Fourier transform of the probability density function
In probability theory and statistics, the characteristic function of any real-valued random variable completely defines its probability distribution. If
Characteristic function (probability theory)
Characteristic_function_(probability_theory)
Set of methods for supervised statistical learning
In machine learning, a support vector machine (SVM) or support vector network is a supervised max-margin model with associated learning algorithms that
Support_vector_machine
Set of vectors used to define coordinates
random vectors are with high probability almost orthogonal, and the number of independent random vectors, which all are with given high probability pairwise
Basis_(linear_algebra)
Average uncertainty in variable's states
the probability vector p 1 , … , p n {\displaystyle p_{1},\ldots ,p_{n}} . It is worth noting that if we drop the "small for small probabilities" property
Entropy_(information_theory)
Preorder on vectors of real numbers
general-length probability vectors: the singleton vector majorizes all other probability vectors, and the uniform distribution is majorized by all probability vectors
Majorization
Machine learning calibration technique
classification model into a probability distribution over classes. The method was invented by John Platt in the context of support vector machines, replacing
Platt_scaling
Situation where total gains match total losses
value of the game. Multiplying u by that value gives a probability vector, giving the probability that the maximizing player will choose each possible pure
Zero-sum_game
Probability distribution
distribution to D-dimensional probability vectors by taking a logistic transformation of a multivariate normal distribution. The probability density function is:
Logit-normal_distribution
Statistical concept
ϕ i = 1 … K = mixture weight, i.e., prior probability of a particular component i ϕ = K -dimensional vector composed of all the individual ϕ 1 … K ;
Mixture_model
Concept in statistics
probability and statistics, a compound probability distribution (also known as a mixture distribution or contagious distribution) is the probability distribution
Compound probability distribution
Compound_probability_distribution
Algorithm used by Google Search to rank web pages
columns with only zero values, they should be replaced with the initial probability vector P {\displaystyle \mathbf {P} } . In other words, M ′ := M + D {\displaystyle
PageRank
Concept in probability theory
identically distributed and normally distributed. If the probability distribution of a vector-valued random variable X = ( X1, ..., Xn )T is the same as
Maxwell's_theorem
Measure of distinguishability between two quantum states
divergence of the probability vector ( λ 1 , … , λ n ) {\displaystyle (\lambda _{1},\ldots ,\lambda _{n})} with respect to the probability vector ( μ 1 , …
Quantum_relative_entropy
Measure for evaluating probabilistic forecasts
or algorithm will return a probability vector p ∈ [ 0 , 1 ] m {\displaystyle \mathbf {p} \in [0,1]^{m}} with probabilities for each of the m {\displaystyle
Scoring_rule
Examples of the probabilistic construct
probability, the process satisfies the Markov property. The PageRank of a specific website is simply its probability value in the steady-state vector
Examples_of_Markov_chains
Interpretation of probability
Bayesian probability (/ˈbeɪziən/ BAY-zee-ən or /ˈbeɪʒən/ BAY-zhən) is an interpretation of the concept of probability, in which, instead of frequency or
Bayesian_probability
Mathematical model for sequential decision making under uncertainty
αr } of possible outputs, or actions, with r ≤ s, an initial state probability vector p(0) = ≪ p1(0), ..., ps(0) ≫, a computable function A which after
Markov_decision_process
Probability theory and statistics have some commonly used conventions, in addition to standard mathematical notation and mathematical symbols. Random
Notation in probability and statistics
Notation_in_probability_and_statistics
Vector in relativity
In special relativity, a four-vector (or 4-vector, sometimes Lorentz vector) is an element of a four-dimensional vector space object with four components
Four-vector
estimation multivariate random variable A vector whose components are random variables on the same probability space. mutual exclusivity mutual independence
Glossary of probability and statistics
Glossary_of_probability_and_statistics
Hungarian and American mathematician and physicist (1903–1957)
In this model, the (transposed) probability vector p represents the prices of the goods while the probability vector q represents the "intensity" at which
John_von_Neumann
Mathematical entity to describe the probability of each possible measurement on a system
with the probabilities ps that the quantum is in those states. States can be formulated in terms of observables, rather than as vectors in a vector space
Quantum_state
Computer hardware technology that uses quantum mechanics
quantum state vector behaves similarly to a (classical) probability vector, with one key difference: unlike probabilities, probability amplitudes are
Quantum_computing
Type of probability distribution
on the same probability space, the multivariate or joint probability distribution for X , Y , … {\displaystyle X,Y,\ldots } is a probability distribution
Joint probability distribution
Joint_probability_distribution
Collection of random variables
In probability theory and related fields a stochastic (/stəˈkæstɪk/) or random process is a mathematical object usually defined as a family of random
Stochastic_process
Ballistics measure of a weapon system's precision
Circular error probable (CEP), also circular error probability or circle of equal probability, is a measure of a weapon system's precision in the military
Circular_error_probable
Probability concept
\mathbb {R} ^{n\times n}} which is induced by a scalar product, and any probability vector π {\displaystyle \pi } , there exists a unique transition matrix P
Discrete-time_Markov_chain
Monte Carlo algorithm
where the all-zeros vector occurs with probability 1/2, and all other vectors are equally probable, and so have a probability of 1 2 ( 2 100 − 1 )
Gibbs_sampling
Mathematical concept applicable to physics
in applied mathematics and vector calculus which has many applications in physics. For transport phenomena, flux is a vector quantity, describing the magnitude
Flux
Theorem in linear algebra
} Donsker–Varadhan–Friedland formula: Let p be a probability vector and x a strictly positive vector. Then, r = sup p inf x > 0 ∑ i = 1 n p i [ A x ]
Perron–Frobenius_theorem
Length in a vector space
In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance
Norm_(mathematics)
Calculus of vector-valued functions
Vector calculus or vector analysis is a branch of mathematics concerned with the differentiation and integration of vector fields, primarily in three-dimensional
Vector_calculus
Quantum analog of probabilistic automata
transition matrices, and a probability vector for the state; this gives a probabilistic finite automaton. The entries in the state vector must be real numbers
Quantum_finite_automaton
Probability concept
_{\geq 0}\to S} . A continuous-time Markov chain is defined by: A probability vector λ {\displaystyle \lambda } on S {\displaystyle S} (which below we
Continuous-time_Markov_chain
Variable representing a random phenomenon
optionally be represented as a vector of real-valued random variables (all defined on the same underlying probability space Ω {\displaystyle \Omega }
Random_variable
Family of stochastic optimization methods
single vector of four probabilities (p1, p2, p3, p4) where each component of p defines the probability of that position being a 1. Using this probability vector
Estimation of distribution algorithm
Estimation_of_distribution_algorithm
Estimate of an unobservable underlying probability density function
In statistics, probability density estimation or simply density estimation is the construction of an estimate, based on observed data, of an unobservable
Density_estimation
Technique in information theory
P {\displaystyle P\,} as a Markov state transition probability matrix, the vector of probabilities of the 'states' after t {\displaystyle t\,} steps,
Information_bottleneck_method
When the occurrence of one event does not affect the likelihood of another
Independence is a fundamental notion in probability theory, as in statistics and the theory of stochastic processes. Two events are independent, statistically
Independence (probability theory)
Independence_(probability_theory)
Decision theory term
R={\begin{bmatrix}1500&300&-800\\900&600&-200\\500&500&500\end{bmatrix}}} The probability vector is: p = [ 0.5 0.3 0.2 ] {\displaystyle p={\begin{bmatrix}0.5\\0.3\\0
Expected value of perfect information
Expected_value_of_perfect_information
Measure of covariance of components of a random vector
(2010). "Lectures on probability theory and mathematical statistics". Eaton, Morris L. (1983). Multivariate Statistics: a Vector Space Approach. John
Covariance_matrix
Classical quantization technique from signal processing
Vector quantization (VQ) is a classical quantization technique from signal processing that allows the modeling of probability density functions by the
Vector_quantization
Probability distribution
{\alpha }})} , is a family of continuous multivariate probability distributions parameterized by a vector α of positive reals. It is a multivariate generalization
Dirichlet_distribution
Error in statistical reasoning with groups
Simpson's paradox is a phenomenon in probability and statistics in which a trend appears in several groups of data but disappears or reverses when the
Simpson's_paradox
Machine learning problem
predicted probability or score (such as a distorted probability distribution or the "signed distance to the hyperplane" in a support vector machine).
Probabilistic_classification
Branch of applied mathematics
the (transposed) probability vector p → {\displaystyle {\vec {p}}} represents the prices of the goods, while the probability vector q → {\displaystyle
Mathematical_economics
Measure of the joint variability
In probability theory and statistics, covariance is a measure of the joint variability of two random variables. The sign of the covariance shows the tendency
Covariance
Type of matrix in probability theory and statistics
between the i-th element of a random vector and j-th element of another random vector. When the two random vectors are the same, the cross-covariance matrix
Cross-covariance_matrix
Algebraic structure in linear algebra
operations of vector addition and scalar multiplication must satisfy certain requirements, called vector axioms. Real vector spaces and complex vector spaces
Vector_space
Statistical Markov model
in the form of a high-dimensional vector, is used as a conditioning variable of the HMM state transition probabilities. Under such a setup, eventually is
Hidden_Markov_model
Conditional probability used in Bayesian statistics
The posterior probability is a type of conditional probability that results from updating the prior probability with information summarized by the likelihood
Posterior_probability
Matrix representation of a graph
{\textstyle e_{i}} denote the i-th standard basis vector. Then x = e i P {\textstyle x=e_{i}P} is a probability vector representing the distribution of a random
Laplacian_matrix
Equation describing the transport of some quantity
of probability. The chance of finding the particle at some position r and time t flows like a fluid; hence the term probability current, a vector field
Continuity_equation
Value that appears most often in a set of data
is a discrete random variable, the mode is the value x at which the probability mass function P(X) takes its maximum value, i.e., x = argmaxxi P(X =
Mode_(statistics)
Models used to produce word embeddings
technique in natural language processing for obtaining vector representations of words. These vectors capture information about the meaning of the word based
Word2vec
Generative topic model
random mixture of latent topics, and each topic is characterized by a probability distribution over words. The model is a generalization of probabilistic
Latent_Dirichlet_allocation
Concept in probability theory and statistics
In probability theory and statistics, the moment generating function of a real-valued random variable is a generating function that provides an alternative
Moment_generating_function
Inequalities in probability theory
In probability theory, Bernstein inequalities give bounds on the probability that the sum of random variables deviates from its mean. In the simplest
Bernstein inequalities (probability theory)
Bernstein_inequalities_(probability_theory)
Interpretation of quantum mechanics
transformation of the SIC probability vector. As QBism says that quantum theory is an empirically-motivated normative addition to probability theory, Fuchs and
QBism
Smooth approximation of one-hot arg max
linear functions, and the predicted probability for the jth class given a sample tuple x and a weighting vector w is: P ( y = j ∣ x ) = e x T w j ∑ k
Softmax_function
rate matrix Q is, under any invariant probability vector, reversible if and only if its transition probabilities satisfy q j 1 j 2 q j 2 j 3 ⋯ q j n −
Kolmogorov's_criterion
Term in statistical hypothesis testing
In frequentist statistics, power is the probability of detecting an effect (i.e. rejecting the null hypothesis) given that some prespecified effect actually
Power_(statistics)
Computational concept
n-k} bits of the input r {\displaystyle r} to any fixed values, the probability vector p {\displaystyle p} of the output f ( r ) {\displaystyle f(r)} over
Randomness_extractor
Interpretation of probability
Frequentist probability or frequentism is an interpretation of probability; it defines an event's probability (the long-run probability) as the limit
Frequentist_probability
Array of numbers
equation. Stochastic matrices are square matrices whose rows are probability vectors, that is, whose entries are non-negative and sum up to one. Stochastic
Matrix_(mathematics)
Notions of probabilistic convergence, applied to estimation and asymptotic analysis
In probability theory, there exist several different notions of convergence of sequences of random variables, including convergence in probability, convergence
Convergence of random variables
Convergence_of_random_variables
Aspect of probability and statistics
In probability theory and statistics, the marginal distribution of a subset of a collection of random variables is the probability distribution of the
Marginal_distribution
Sorting method in information retrieval
which has the highest scores or most relevant to query vector. In probabilistic model, probability theory has been used as a principal means for modeling
Ranking (information retrieval)
Ranking_(information_retrieval)
Similarity measure for number sequences
between two non-zero vectors defined in an inner product space. Cosine similarity is the cosine of the angle between the vectors; that is, it is the dot
Cosine_similarity
Mathematical theorem
the vector x n = ( x 1 , x 2 , … , x n ) {\displaystyle x^{n}=(x_{1},x_{2},\ldots ,x_{n})} . Then, we have the following bound on the probability that
Sanov's_theorem
Type of shift space studied in ergodic theory
{\displaystyle \sum _{j=1}^{n}p_{ij}=1} for all i. The stationary probability vector π = (πi) has all πi ≥ 0, ∑ π i = 1 {\textstyle \sum \pi _{i}=1} and
Subshift_of_finite_type
Probability distribution
In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution for nonnegative-valued random variables. Up to
Rayleigh_distribution
Distribution of an uncertain quantity
A prior probability distribution (often simply called the prior probability, prior distribution, or prior) of an uncertain quantity is its assumed probability
Prior_probability
In probability theory and statistics, a complex random vector is typically a tuple of complex-valued random variables, and generally is a random variable
Complex_random_vector
Mathematical description of quantum state
interpretation of quantum mechanics, the Born rule, relating transition probabilities to inner products. The Schrödinger equation determines how wave functions
Wave_function
Field of study among musicians and computer scientists
become note or pitch values, and a probability vector for each note is constructed, completing a transition probability matrix (see below). An algorithm
Pop_music_automation
Measure of the accuracy of probabilistic predictions
original (PDF) on 2017-10-23. Murphy, A. H. (1973). "A new vector partition of the probability score". Journal of Applied Meteorology. 12 (4): 595–600.
Brier_score
Concepts from linear algebra
algebra, an eigenvector (/ˈaɪɡən-/ EYE-gən-) or characteristic vector is a (nonzero) vector that has its direction unchanged (or reversed) by a given linear
Eigenvalues_and_eigenvectors
Probability that random variable X is less than or equal to x
In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X {\displaystyle X} , or just distribution
Cumulative distribution function
Cumulative_distribution_function
Statistical model for a binary dependent variable
outcome y will be in category y=n, conditional on the vector of covariates x. The sum of these probabilities over all categories must equal 1. Using the mathematically
Logistic_regression
Theorem in quantum mechanics
these probabilities sum to 1 for any set of unit vectors comprising an orthonormal basis. Moreover, the probability associated with a unit vector is a
Gleason's_theorem
American economist
is a way of representing a three dimensional probability vector in a two dimensional space. The probability of a given outcome is denoted by a euclidean
Mark_J._Machina
Probability distribution
distributions). (For other names, see Naming.) The univariate probability distribution is generalized for vectors in the multivariate normal distribution and for matrices
Normal_distribution
Area of functional analysis and convex analysis
metrizable subset C of a locally convex topological vector space V, given c in C there exists a probability measure w on C supported on the set E of extreme
Choquet_theory
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