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Markov's principle (also known as the Leningrad principle [1]), named after Andrey Markov Jr, is a conditional existence statement for which there are many equivalent formulations, as discussed below. The principle is logically valid classically, but not in intuitionistic constructive mathematics. However, many particular instances of it are ...
In the presence of Markov's principle, the syntactical restrictions may be somewhat loosened. [ 1 ] When considering the domain of all numbers (e.g. when taking ψ ( x ) {\displaystyle \psi (x)} to be the trivial x = x {\displaystyle x=x} ), the above reduces to the previous form of Church's thesis.
In probability theory, Markov's inequality gives an upper bound on the probability that a non-negative random variable is greater than or equal to some positive constant. Markov's inequality is tight in the sense that for each chosen positive constant, there exists a random variable such that the inequality is in fact an equality.
Andrey Markov, Sr., invented the Markov chains, proved Markov brothers' inequality, author of the hidden Markov model, Markov number, Markov property, Markov's inequality, Markov processes, Markov random field, Markov algorithm etc. Andrey Markov, Jr., author of Markov's principle and Markov's rule in logics
The related Causal Markov (CM) condition states that, conditional on the set of all its direct causes, a node is independent of all variables which are not effects or direct causes of that node. [3] In the event that the structure of a Bayesian network accurately depicts causality , the two conditions are equivalent.
The base logic of constructive analysis is intuitionistic logic, which means that the principle of excluded middle is not automatically assumed for every proposition.If a proposition . is provable, this exactly means that the non-existence claim . being provable would be absurd, and so the latter cannot also be provable in a consistent theory.
In probability theory and ergodic theory, a Markov operator is an operator on a certain function space that conserves the mass (the so-called Markov property). If the underlying measurable space is topologically sufficiently rich enough, then the Markov operator admits a kernel representation. Markov operators can be linear or non-linear.
Every adapted right continuous Feller process on a filtered probability space (,, ()) satisfies the strong Markov property with respect to the filtration (+), i.e., for each (+)-stopping time, conditioned on the event {<}, we have that for each , + is independent of + given .