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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.
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
In probability theory and information theory, the mutual information (MI) of two random variables is a measure of the mutual dependence between the two variables. More specifically, it quantifies the " amount of information " (in units such as shannons ( bits ), nats or hartleys ) obtained about one random variable by observing the other random ...
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:
The above definition is for discrete random variables. The continuous version of discrete conditional entropy is called conditional differential (or continuous) entropy . Let X {\displaystyle X} and Y {\displaystyle Y} be a continuous random variables with a joint probability density function f ( x , y ) {\displaystyle f(x,y)} .
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.
In statistics, probability theory and information theory, pointwise mutual information (PMI), [1] or point mutual information, is a measure of association.It compares the probability of two events occurring together to what this probability would be if the events were independent.
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, ′ = ′ = (′) ′.