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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.
When two or more random variables are defined on a probability space, it is useful to describe how they vary together; that is, it is useful to measure the relationship between the variables. A common measure of the relationship between two random variables is the covariance.
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:
In probability theory and statistics, a Markov chain or Markov process is a stochastic process ... of random variables X 1, X 2, ... homogeneous by Bayes' rule.
In mathematical statistics, the Fisher information is a way of measuring the amount of information that an observable random variable X carries about an unknown parameter θ of a distribution that models X. Formally, it is the variance of the score, or the expected value of the observed information.
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
The support of a random variable is defined to be the topological support of this measure, i.e. =. Now we can formally define the conditional probability measure given the value of one (or, via the product topology , more) of the random variables.
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.