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In probability theory, the joint probability distribution is the probability distribution of all possible pairs of outputs of two random variables that are defined on the same probability space. The joint distribution can just as well be considered for any given number of random variables.
Formally, an exchangeable sequence of random variables is a finite or infinite sequence X 1, X 2, X 3, ... of random variables such that for any finite permutation σ of the indices 1, 2, 3, ..., (the permutation acts on only finitely many indices, with the rest fixed), the joint probability distribution of the permuted sequence
Also confidence coefficient. A number indicating the probability that the confidence interval (range) captures the true population mean. For example, a confidence interval with a 95% confidence level has a 95% chance of capturing the population mean. Technically, this means that, if the experiment were repeated many times, 95% of the CIs computed at this level would contain the true population ...
One can compute this directly, without using a probability distribution (distribution-free classifier); one can estimate the probability of a label given an observation, (| =) (discriminative model), and base classification on that; or one can estimate the joint distribution (,) (generative model), from that compute the conditional probability ...
The i.i.d. assumption is also used in the central limit theorem, which states that the probability distribution of the sum (or average) of i.i.d. variables with finite variance approaches a normal distribution. [4] The i.i.d. assumption frequently arises in the context of sequences of random variables. Then, "independent and identically ...
A discrete probability distribution is the probability distribution of a random variable that can take on only a countable number of values [15] (almost surely) [16] which means that the probability of any event can be expressed as a (finite or countably infinite) sum: = (=), where is a countable set with () =.
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.
where is the Kullback–Leibler divergence, and is the outer product distribution which assigns probability () to each (,).. Notice, as per property of the Kullback–Leibler divergence, that (;) is equal to zero precisely when the joint distribution coincides with the product of the marginals, i.e. when and are independent (and hence observing tells you nothing about ).