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In probability theory, the chain rule [1] (also called the general product rule [2] [3]) describes how to calculate the probability of the intersection of, not necessarily independent, events or the joint distribution of random variables respectively, using conditional probabilities.
In probability theory and statistics, a Markov chain or Markov process is a stochastic process describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event.
In this situation, the chain rule represents the fact that the derivative of f ∘ g is the composite of the derivative of f and the derivative of g. This theorem is an immediate consequence of the higher dimensional chain rule given above, and it has exactly the same formula. The chain rule is also valid for Fréchet derivatives in Banach spaces.
The chain rule [citation needed] for Kolmogorov complexity is an analogue of the chain rule for information entropy, which states: (,) = + (|)That is, the combined randomness of two sequences X and Y is the sum of the randomness of X plus whatever randomness is left in Y once we know X.
In probability theory, ... This identity is known as the chain rule of probability. Since these are probabilities, in the two-variable case
Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations , probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms .
Chain rule for Kolmogorov complexity; Challenge–dechallenge–rechallenge; Champernowne distribution; Change detection. Change detection (GIS) Chapman–Kolmogorov equation; Chapman–Robbins bound; Characteristic function (probability theory) Chauvenet's criterion; Chebyshev center; Chebyshev's inequality
The chain rule [20] for Kolmogorov complexity states that there exists a constant c such that for all X and Y: K(X,Y) = K(X) + K(Y|X) + c*max(1,log(K(X,Y))). It states that the shortest program that reproduces X and Y is no more than a logarithmic term larger than a program to reproduce X and a program to reproduce Y given X.