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Bayesian statistics are based on a different philosophical approach for proof of inference.The mathematical formula for Bayes's theorem is: [|] = [|] [] []The formula is read as the probability of the parameter (or hypothesis =h, as used in the notation on axioms) “given” the data (or empirical observation), where the horizontal bar refers to "given".
Proof complexity, computational resources required to prove statements; Proof procedure, method for producing proofs in proof theory; Proof theory, a branch of mathematical logic that represents proofs as formal mathematical objects; Statistical proof, demonstration of degree of certainty for a hypothesis
Surrogate data testing [1] (or the method of surrogate data) is a statistical proof by contradiction technique similar to permutation tests [2] and parametric bootstrapping.It is used to detect non-linearity in a time series. [3]
The Italian mathematician Gerolamo Cardano (1501–1576) stated without proof that the accuracies of empirical statistics tend to improve with the number of trials. [9] [3] This was then formalized as a law of large numbers. A special form of the LLN (for a binary random variable) was first proved by Jacob Bernoulli.
Maximum likelihood estimation is a generic technique for estimating the unknown parameters in a statistical model by constructing a log-likelihood function corresponding to the joint distribution of the data, then maximizing this function over all possible parameter values. In order to apply this method, we have to make an assumption about the ...
The expression "statistical proof" may be used technically or colloquially in areas of pure mathematics, such as involving cryptography, chaotic series, and probabilistic number theory or analytic number theory. [23] [24] [25] It is less commonly used to refer to a mathematical proof in the branch of mathematics known as mathematical statistics.
Statistics subsequently branched out into various directions, including decision theory, Bayesian statistics, exploratory data analysis, robust statistics, and non-parametric statistics. Neyman-Pearson hypothesis testing made significant contributions to decision theory, which is widely employed, particularly in statistical quality control.
The total variation distance (or half the norm) arises as the optimal transportation cost, when the cost function is (,) =, that is, ‖ ‖ = (,) = {(): =, =} = [], where the expectation is taken with respect to the probability measure on the space where (,) lives, and the infimum is taken over all such with marginals and , respectively.