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Let : be a continuously-differentiable, strictly convex function defined on a convex set. The Bregman distance associated with F for points p , q ∈ Ω {\displaystyle p,q\in \Omega } is the difference between the value of F at point p and the value of the first-order Taylor expansion of F around point q evaluated at point p :
The study of dynamic equations on time scales reveals such discrepancies, and helps avoid proving results twice—once for differential equations and once again for difference equations. The general idea is to prove a result for a dynamic equation where the domain of the unknown function is a so-called time scale (also known as a time-set ...
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 , nats or hartleys) obtained about one random variable by observing the other random variable.
In essence probability is influenced by a person's information about the possible occurrence of an event. For example, let the event be 'I have a new phone'; event be 'I have a new watch'; and event be 'I am happy'; and suppose that having either a new phone or a new watch increases the probability of my being happy.
The two most important classes of divergences are the f-divergences and Bregman divergences; however, other types of divergence functions are also encountered in the literature. The only divergence for probabilities over a finite alphabet that is both an f -divergence and a Bregman divergence is the Kullback–Leibler divergence. [ 8 ]
The least squares regression line is a method in simple linear regression for modeling the linear relationship between two variables, and it serves as a tool for making predictions based on new values of the independent variable. The calculation is based on the method of the least squares criterion. The goal is to minimize the sum of the ...
Differentiation with respect to time or one of the other variables requires application of the chain rule, [1] since most problems involve several variables. Fundamentally, if a function F {\displaystyle F} is defined such that F = f ( x ) {\displaystyle F=f(x)} , then the derivative of the function F {\displaystyle F} can be taken with respect ...
It has a basis in one-to-one correspondence with the set of k-simplices in . To define a basis explicitly, one has to choose an orientation of each simplex. One standard way to do this is to choose an ordering of all the vertices and give each simplex the orientation corresponding to the induced ordering of its vertices.