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Constantin Carathéodory's alternative definition of the differentiability of a function can be used to give an elegant proof of the chain rule. [6] Under this definition, a function f is differentiable at a point a if and only if there is a function q, continuous at a and such that f(x) − f(a) = q(x)(x − a).
This rule allows one to express a joint probability in terms of only conditional probabilities. [4] The rule is notably used in the context of discrete stochastic processes and in applications, e.g. the study of Bayesian networks, which describe a probability distribution in terms of conditional probabilities.
As in the discrete case there is a chain rule for differential entropy: (|) = (,) [3]: 253 Notice however that this rule may not be true if the involved differential entropies do not exist or are infinite.
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
The data processing inequality is closely related to conditional mutual information and can be proven using the chain rule. Interaction information The ...
In propositional logic, hypothetical syllogism is the name of a valid rule of inference (often abbreviated HS and sometimes also called the chain argument, chain rule, or the principle of transitivity of implication). The rule may be stated:
In addition, the package contains 12 bills for consideration under a closed rule – meaning no amendments can be proposed – that fulfill multiple Republican objectives, such as requiring proof ...
Chain rule Suppose that f : A → R is a real-valued function defined on a subset A of R n, and that f is differentiable at a point a. There are two forms of the chain rule applying to the gradient. First, suppose that the function g is a parametric curve; that is, a function g : I → R n maps a subset I ⊂ R into R n.