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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 calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions f and g in terms of the derivatives of f and g.More precisely, if = is the function such that () = (()) for every x, then the chain rule is, in Lagrange's notation, ′ = ′ (()) ′ (). or, equivalently, ′ = ′ = (′) ′.
The generalization of the preceding two-variable case is the joint probability distribution of ... This identity is known as the chain rule of probability.
2.5 Chain rule. 2.6 Dot product rule. 2.7 Cross product rule. 3 Second derivative identities. ... We have the following special cases of the multi-variable chain rule.
The chain rule has a particularly elegant statement in terms of total derivatives. It says that, for two functions f {\displaystyle f} and g {\displaystyle g} , the total derivative of the composite function f ∘ g {\displaystyle f\circ g} at a {\displaystyle a} satisfies
In contrast to the conditional entropy for discrete random variables, the conditional differential entropy may be negative. As in the discrete case there is a chain rule for differential entropy: (|) = (,) [3]: 253
This property takes two different forms respectively for functions of one and several complex variables: for the n > 1 case, to express the chain rule in its full generality it is necessary to consider two domains ′ and ″ and two maps: ′ and : ″ having natural smoothness requirements.
As in the discrete case the joint differential entropy of a set of random variables is smaller or equal than the sum of the entropies of the individual random variables: (,, …,) = [3]: 253 The following chain rule holds for two random variables: