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Derivation of the laws of probability theory
Cox's theorem, named after the physicist Richard Threlkeld Cox, is a derivation of the laws of probability theory from a certain set of postulates. This
Cox's_theorem
American physicist (1898–1991)
Richard Threlkeld Cox (August 5, 1898 – May 2, 1991) was a professor of physics at Johns Hopkins University, known for Cox's theorem relating to the foundations
Richard_Threlkeld_Cox
Mathematical rule for inverting probabilities
In probability theory, Bayes' theorem (alternatively Bayes' law or Bayes' rule), named after Thomas Bayes (/beɪz/), gives a mathematical rule for inverting
Bayes'_theorem
Foundations of probability theory
arguing that the axioms are satisfied by this definition. For example, Cox's theorem derives the laws of probability based on a "logical" definition of probability
Probability_axioms
theorem (logic) Diaconescu's theorem (mathematical logic) Easton's theorem (set theory) Erdős–Dushnik–Miller theorem (set theory) Erdős–Rado theorem (set
List_of_theorems
17th-century conjecture proved by Andrew Wiles in 1994
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that there are no positive integers a
Fermat's_Last_Theorem
Results about asymptotic posterior normality
In Bayesian inference, the Bernstein–von Mises theorem provides the basis for using Bayesian credible sets for confidence statements in parametric models
Bernstein–von_Mises_theorem
Theory and paradigm of statistics
Bayesian statistical methods use Bayes' theorem to compute and update probabilities after obtaining new data. Bayes' theorem describes the conditional probability
Bayesian_statistics
Interpretation of probability
arguments, such as Cox axioms, the Dutch book argument, arguments based on decision theory and de Finetti's theorem. Richard T. Cox showed that Bayesian
Bayesian_probability
likelihood Bayesian probability Principle of indifference Credal set Cox's theorem Principle of maximum entropy Information entropy Urn problems Extractor
List_of_probability_topics
Fundamental theorem in probability theory and statistics
In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the distribution of a normalized version of the sample
Central_limit_theorem
Calculation of complex statistical distributions
(Ergodic Theorem). And we need aperiodicity, irreducibility and extra conditions such as reversibility to ensure the Central Limit Theorem holds in MCMC
Markov_chain_Monte_Carlo
Statistical model written in multiple levels
method. The sub-models combine to form the hierarchical model, and Bayes' theorem is used to integrate them with the observed data and account for all the
Bayesian hierarchical modeling
Bayesian_hierarchical_modeling
Principle in Bayesian statistics
and is conventionally called the partition function. (The Pitman–Koopman theorem states that the necessary and sufficient condition for a sampling distribution
Principle_of_maximum_entropy
Number measuring the chance an event occurs
interpreted as events and probability as a measure on a class of sets. In Cox's theorem, probability is taken as a primitive (i.e., not further analyzed), and
Probability
Classification algorithm in statistics
where the second line was derived through Fubini's theorem Notice that R ( h ) {\displaystyle R(h)} is minimised by taking ∀ x ∈ X
Bayes_classifier
Applications of logic under uncertainty
learning Bayesian inference, Bayesian network, Bayesian probability Cox's theorem Fréchet inequalities Imprecise probability Non-monotonic logic Possibility
Probabilistic_logic
Method of statistical inference
/ˈbeɪʒən/ BAY-zhən) is a method of statistical inference in which Bayes' theorem is used to calculate a probability of a hypothesis, given prior evidence
Bayesian_inference
Probabilistic theory of knowledge
method in contrast to traditional epistemology is that its concepts and theorems can be defined with a high degree of precision. The most characteristic
Bayesian_epistemology
Probability rule of thumb
prior probability assigned to a hypothesis is 0 or 1, then, by Bayes' theorem, the posterior probability (probability of the hypothesis, given the evidence)
Cromwell's_rule
Monte Carlo algorithm
Bayesian inference Bayesian probability Bayes' theorem Bernstein–von Mises theorem Coherence Cox's theorem Cromwell's rule Likelihood principle Principle
Gibbs_sampling
Conditional probability used in Bayesian statistics
this student is a girl? The correct answer can be computed using Bayes' theorem. The event G is that the student observed is a girl, and the event T is
Posterior_probability
Lower bound on the log-likelihood of some observed data
Bayesian inference Bayesian probability Bayes' theorem Bernstein–von Mises theorem Coherence Cox's theorem Cromwell's rule Likelihood principle Principle
Evidence_lower_bound
matrix Covariance function Covariate Cover's theorem Coverage probability Cox process Cox's theorem Cox–Ingersoll–Ross model Cramér–Rao bound Cramér–von
List_of_statistics_articles
Concept in probability theory
{\displaystyle \theta } are the parameters of the model. Using Bayes' theorem we can expand p ( θ | x ) = p ( x | θ ) p ( θ ) p ( x ) , {\displaystyle
Conjugate_prior
In Bayesian probability theory
Bayesian inference Bayesian probability Bayes' theorem Bernstein–von Mises theorem Coherence Cox's theorem Cromwell's rule Likelihood principle Principle
Marginal_likelihood
Statistics concept
incomplete/uncertain knowledge. The defense of probability is mainly based on Cox's theorem, which starts from four postulates concerning rational reasoning in
Bayesian_programming
Criterion for model selection
Bayesian inference Bayesian probability Bayes' theorem Bernstein–von Mises theorem Coherence Cox's theorem Cromwell's rule Likelihood principle Principle
Bayesian information criterion
Bayesian_information_criterion
Function related to statistics and probability theory
{\text{HH}})=0.25} , a conclusion which could only be reached via Bayes' theorem given knowledge about the marginal probabilities P ( p H = 0.5 ) {\textstyle
Likelihood_function
Analytical expression in statistics
information. The approximation is justified by the Bernstein–von Mises theorem, which states that, under regularity conditions, the error of the approximation
Laplace's_approximation
Probabilistic graphical representation of causal relationships
network can thus be considered a mechanism for automatically applying Bayes' theorem to complex problems. The most common exact inference methods are: variable
Bayesian_network
In probability theory, a rule for assigning epistemic probabilities
parameterization the probability density will be uniform. Liouville's theorem justifies the use of canonically conjugate variables, such as positions
Principle_of_indifference
Type of "good" decision rule in Bayesian statistics
\!} and F ( x ∣ θ ) {\displaystyle F(x\mid \theta )\,\!} using Bayes' theorem). Having made explicit the expected loss for each given x {\displaystyle
Admissible_decision_rule
Concept in Bayesian statistics
Bayesian inference Bayesian probability Bayes' theorem Bernstein–von Mises theorem Coherence Cox's theorem Cromwell's rule Likelihood principle Principle
Credible_interval
Thought experiment, to justify Bayesian probability
choice theory Mathematics of bookmaking Von Neumann-Morgenstern utility theorem Scoring rule Bovens, Luc; Hartmann, Stephan (2003). "Coherence". Bayesian
Dutch_book_arguments
Ratio of competing statistical models
{\displaystyle \Pr(M|D)} of a model M given data D is given by Bayes' theorem: Pr ( M | D ) = Pr ( D | M ) Pr ( M ) Pr ( D ) . {\displaystyle \Pr(M|D)={\frac
Bayes_factor
Bayesian statistical inference method
summarised by the hyperparameters η {\displaystyle \eta \;} . Using Bayes' theorem, p ( θ ∣ y ) = p ( y ∣ θ ) p ( θ ) p ( y ) = p ( y ∣ θ ) p ( y ) ∫ p (
Empirical_Bayes_method
Bayesian inference Bayesian probability Bayes' theorem Bernstein–von Mises theorem Coherence Cox's theorem Cromwell's rule Likelihood principle Principle
Hyperprior
Theorem in political science
In political science and social choice, Black's median voter theorem says that if voters and candidates are distributed along a one-dimensional political
Median_voter_theorem
Method of estimating the parameters of a statistical model
calculate the posterior density of θ {\displaystyle \theta } using Bayes' theorem: θ ↦ f ( θ ∣ x ) = f ( x ∣ θ ) g ( θ ) ∫ Θ f ( x ∣ ϑ ) g ( ϑ ) d ϑ {\displaystyle
Maximum a posteriori estimation
Maximum_a_posteriori_estimation
Distribution of new data marginalized over the posterior
Bayesian inference Bayesian probability Bayes' theorem Bernstein–von Mises theorem Coherence Cox's theorem Cromwell's rule Likelihood principle Principle
Posterior predictive distribution
Posterior_predictive_distribution
Parameter of a prior distribution in Bayesian statistics
Bayesian inference Bayesian probability Bayes' theorem Bernstein–von Mises theorem Coherence Cox's theorem Cromwell's rule Likelihood principle Principle
Hyperparameter (Bayesian statistics)
Hyperparameter_(Bayesian_statistics)
Mathematical decision rule
}}.} This is a definition, and not an application of Bayes' theorem, since Bayes' theorem can only be applied when all distributions are proper. However
Bayes_estimator
Polynomial ideals are finitely generated
fundamental theorems on polynomials, the Nullstellensatz (zero-locus theorem) and the syzygy theorem (theorem on relations). These three theorems were the
Hilbert's_basis_theorem
Method of statistical analysis
parameters is combined with the data's likelihood function according to Bayes' theorem to yield the posterior belief about the parameters β {\displaystyle {\boldsymbol
Bayesian_linear_regression
Distribution of an uncertain quantity
distribution on the interval [0, 1]. This is obtained by applying Bayes' theorem to the data set consisting of one observation of dissolving and one of
Prior_probability
Proposition in statistics
1(2), pp.75-78] Evans, Michael (2013). "What does the proof of Birnbaum's theorem prove?". arXiv:1302.5468 [math.ST]. Mayo, D. (2010). "An error in the argument
Likelihood_principle
Mathematical framework to model epistemic uncertainty
meaningful sense. For example, DS theory violates the requirements for Cox's theorem, which implies that it cannot be considered a coherent (contradiction-free)
Dempster–Shafer_theory
On the number of common zeros of Laurent polynomials
The Bernstein–Kushnirenko theorem, also called Bernstein–Khovanskii–Kushnirenko theorem (BKK theorem), states that the number of nonzero complex solutions
Bernstein–Kushnirenko_theorem
Number divisible only by 1 and itself
than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself
Prime_number
Method for numerical integration
distributions. It was developed in 2004 by physicist John Skilling. Bayes' theorem can be used for model selection, where one has a pair of competing models
Nested_sampling_algorithm
Functional equation characterizing associative binary operations
generators and ordinal sums. In the foundations of Bayesian probability and Cox's theorem, one considers a real-valued plausibility measure on propositions and
Associativity_equation
Mathematical methods used in Bayesian inference and machine learning
bound on the log-evidence of the data. By the generalized Pythagorean theorem of Bregman divergence, of which KL-divergence is a special case, it can
Variational_Bayesian_methods
Condition under which an odd prime is a sum of two squares
In additive number theory, Fermat's theorem on sums of two squares states that an odd prime p can be expressed as: p = x 2 + y 2 , {\displaystyle p=x^{2}+y^{2}
Fermat's theorem on sums of two squares
Fermat's_theorem_on_sums_of_two_squares
Statistical theorem
In statistics, Wilks' theorem offers an asymptotic distribution of the log-likelihood ratio statistic, which can be used to produce confidence intervals
Wilks'_theorem
Analog of Pareto efficiency for situations with incomplete information
Bayesian inference Bayesian probability Bayes' theorem Bernstein–von Mises theorem Coherence Cox's theorem Cromwell's rule Likelihood principle Principle
Bayesian_efficiency
theory started, leading to the receiver operating characteristic 1946 – Cox's theorem derives the axioms of probability from simple logical assumptions, 1948
Timeline of probability and statistics
Timeline_of_probability_and_statistics
Experimental design framework
{\displaystyle \xi } , the posterior probability can be calculated using Bayes' theorem p ( θ ∣ y , ξ ) = p ( y ∣ θ , ξ ) p ( θ ) p ( y ∣ ξ ) , {\displaystyle
Bayesian_experimental_design
Cotter – chemist and mass spectrometrist Richard Threlkeld Cox – physicist, Cox's theorem Thomas Craig – mathematician Tyler Cymet – physician Maqbool
List of Johns Hopkins University people
List_of_Johns_Hopkins_University_people
Result in social choice theory
The McKelvey–Schofield chaos theorem is a result in social choice theory. It states that if preferences are defined over a multidimensional policy space
McKelvey–Schofield chaos theorem
McKelvey–Schofield_chaos_theorem
Field theory result
In mathematics, Abel's irreducibility theorem, a field theory result described in 1829 by Niels Henrik Abel, asserts that if f(x) is a polynomial over
Abel's_irreducibility_theorem
Methodology for assigning prior probabilities
Bayesian inference Bayesian probability Bayes' theorem Bernstein–von Mises theorem Coherence Cox's theorem Cromwell's rule Likelihood principle Principle
Principle of transformation groups
Principle_of_transformation_groups
Sequence of theorems relating to sets of circles intersecting at a common point
In geometry, Clifford's theorems, named after the English geometer William Kingdon Clifford, are a sequence of theorems relating to intersections of circles
Clifford's_circle_theorems
Field theory theorem
primitive element theorem states that every finite separable field extension is simple, i.e. generated by a single element. This theorem implies in particular
Primitive_element_theorem
Statistical theorem
In statistics, the Rao–Blackwell theorem, sometimes referred to as the Rao–Blackwell–Kolmogorov theorem, is a result that characterizes the transformation
Rao–Blackwell_theorem
Class of statistical survival models
application of the Cox proportional hazards model, sometimes abbreviated to Cox model or to proportional hazards model. However, Cox also noted that biological
Proportional_hazards_model
machine Cox's theorem Equipossible Exotic probability Extractor Free probability Frequency Frequency probability Impossible event Infinite monkey theorem Information
Catalog of articles in probability theory
Catalog_of_articles_in_probability_theory
Theorem in statistics
In statistics, Basu's theorem states that any boundedly complete and sufficient statistic is independent of any ancillary statistic. This is a 1955 result
Basu's_theorem
Conditions under which the congruence x^3 equals p (mod q) is solvable
Cubic reciprocity is a collection of theorems in elementary and algebraic number theory that state conditions under which the congruence x3 ≡ p (mod q)
Cubic_reciprocity
Paradox in expected-utility theory
In microeconomics and decision theory, Rabin's calibration theorem (also known as Rabin's paradox or Rabin's critique) is a theoretical result related
Rabin's_calibration_theorem
Computational method in Bayesian statistics
the ABC context in the method (SMC-ABC). A common incarnation of Bayes' theorem relates the conditional probability (or density) of a particular parameter
Approximate Bayesian computation
Approximate_Bayesian_computation
Statistical principle
on an assumption of the distributional form (see Pitman–Koopman–Darmois theorem below), but remained very important in theoretical work. Roughly, given
Sufficient_statistic
Probability distribution to which random variables or distributions "converge"
a regular parametric model; this is just the central limit theorem. Barndorff-Nielson & Cox provide a direct definition of asymptotic normality: The sequence
Asymptotic_distribution
Theorem of stationary processes
Wold representation theorem (not to be confused with the Wold theorem that is the discrete-time analog of the Wiener–Khinchin theorem), named after Herman
Wold's_theorem
Commutative group (mathematics)
structure theorem for finitely generated modules over a principal ideal domain. In the case of finitely generated abelian groups, this theorem guarantees
Abelian_group
English mathematician
The theorem has been compared to Clifford's circle theorems since they both are an infinite chain of theorems. In 1941 Richmond argued that Cox's chain
Homersham_Cox_(mathematician)
Statistics term
statistic which is not complete. This is important because the Lehmann–Scheffé theorem cannot be applied to such models. Galili and Meilijson 2016 propose the
Completeness_(statistics)
In algebraic geometry, Sumihiro's theorem, introduced by (Sumihiro 1974), states that a normal algebraic variety with an action of a torus can be covered
Sumihiro's_theorem
Representation of a type of random process
{\displaystyle X_{t}} is also a Gaussian process. In other cases, the central limit theorem indicates that X t {\displaystyle X_{t}} will be approximately normally
Autoregressive_model
Probability distribution
which is positive. This is justified by considering the central limit theorem in the log domain (sometimes called Gibrat's law). The log-normal distribution
Log-normal_distribution
History Faculty at the University of Oxford 25 October 2012 Fermat's Last Theorem Marcus du Sautoy, Professor of Mathematics & Simonyi Professor for the
List of In Our Time programmes
List_of_In_Our_Time_programmes
Solution concept of a non-cooperative game
Kakutani fixed-point theorem in his 1950 paper to prove existence of equilibria. His 1951 paper used the simpler Brouwer fixed-point theorem for the same purpose
Nash_equilibrium
Sufficient condition for polynomial irreducibility
the early 20th century, it was also known as the Schönemann–Eisenstein theorem because Theodor Schönemann was the first to publish it. Suppose we have
Eisenstein's_criterion
German polymath and scholar (1777–1855)
Gauss produced the second and third complete proofs of the fundamental theorem of algebra. He also introduced the triple bar symbol (≡) for congruence
Carl_Friedrich_Gauss
Conditions in number theory
Quartic or biquadratic reciprocity is a collection of theorems in elementary and algebraic number theory that state conditions under which the congruence
Quartic_reciprocity
In mathematics, the Chung–Fuchs theorem, named after Chung Kai-lai and Wolfgang Heinrich Johannes Fuchs, states that for a particle undergoing a zero-mean
Chung–Fuchs_theorem
Concept in statistics
uniformly distributed random phase. Where applicable, the central limit theorem dictates that at any point, the sum of these individual plane-wave contributions
Gaussian_random_field
Theorem in statistics
Lehmann–Scheffé theorem provides sufficient conditions for the existence of a best unbiased estimator in a statistical model. The theorem states that any
Lehmann–Scheffé_theorem
Process forming a path from many random steps
approximation theorem. The convergence of a random walk toward the Wiener process is controlled by the central limit theorem, and by Donsker's theorem. For a
Random_walk
Statistical test comparing two probability distributions
two distribution functions across all x values. By the Glivenko–Cantelli theorem, if the sample comes from the distribution F(x), then Dn converges to 0
Kolmogorov–Smirnov_test
Branch of probability theory
properties analogous to the strong law of large numbers and central limit theorem. The renewal function m ( t ) {\displaystyle m(t)} (expected number of
Renewal_theory
German mathematician (1882–1935)
rings, fields, and algebras. She also proved Noether's first and second theorems, which play a fundamental role in mathematical physics, by explaining the
Emmy_Noether
Yuri Matiyasevich completing the theorem in 1970. The theorem is now known as Matiyasevich's theorem or the MRDP theorem. Optimal design In the design of
List of inventions and discoveries by women
List_of_inventions_and_discoveries_by_women
Regular polygon that can be constructed with compass and straightedge
numbers Fm and complete factoring status by Wilfrid Keller. Cox, David A. (2012), "Theorem 10.1.6", Galois Theory, Pure and Applied Mathematics (2nd ed
Constructible_polygon
Correlation of a signal with a time-shifted copy of itself, as a function of shift
{\displaystyle 0} for all other τ {\displaystyle \tau } . The Wiener–Khinchin theorem relates the autocorrelation function R X X {\displaystyle \operatorname
Autocorrelation
Family of functions to transform data
tests. Confidence interval for the Box–Cox transformation can be asymptotically constructed using Wilks's theorem on the profile likelihood function to
Power_transform
Theoretically optimal hypothesis test
1-\beta (\theta )=\operatorname {E} [\varphi (X)|\theta ].} The Karlin–Rubin theorem (named for Samuel Karlin and Herman Rubin) can be regarded as an extension
Uniformly_most_powerful_test
Property of objects which are scaled or mirrored versions of each other
are: the angle bisector theorem, the geometric mean theorem, Ceva's theorem, Menelaus's theorem and the Pythagorean theorem. Similar triangles also provide
Similarity_(geometry)
Type of random mathematical object
process, and this result is sometimes referred to as the mapping theorem. The theorem involves some Poisson point process with mean measure Λ {\displaystyle
Poisson_point_process
Statistical distribution for dependence between random variables
and minimize tail risk and portfolio-optimization applications. Sklar's theorem states that any multivariate joint distribution can be written in terms
Copula_(statistics)
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COXS THEOREM
COXS THEOREM
COXS THEOREM
COXS THEOREM
COXS THEOREM
COXS THEOREM
COXS THEOREM
COXS THEOREM
COXS THEOREM
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