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Independence is a fundamental notion in probability theory, as in statistics and the theory of stochastic processes.Two events are independent, statistically independent, or stochastically independent [1] if, informally speaking, the occurrence of one does not affect the probability of occurrence of the other or, equivalently, does not affect the odds.
In probability theory, conditional independence describes situations wherein an observation is irrelevant or redundant when evaluating the certainty of a hypothesis. . Conditional independence is usually formulated in terms of conditional probability, as a special case where the probability of the hypothesis given the uninformative observation is equal to the probability
k-wise independence has been used in theoretical computer science, where it was used to prove a theorem about the problem MAXEkSAT. k-wise independence is used in the proof that k-independent hashing functions are secure unforgeable message authentication codes.
It is often used in statistics as a tool to prove independence of two statistics, by first demonstrating one is complete sufficient and the other is ancillary, then appealing to the theorem. [2] An example of this is to show that the sample mean and sample variance of a normal distribution are independent statistics, which is done in the ...
In mathematical logic, independence is the unprovability of some specific sentence from some specific set of other sentences. The sentences in this set are referred to as "axioms". The sentences in this set are referred to as "axioms".
An axiom P is independent if there are no other axioms Q such that Q implies P. . In many cases independence is desired, either to reach the conclusion of a reduced set of axioms, or to be able to replace an independent axiom to create a more concise system (for example, the parallel postulate is independent of other axioms of Euclidean geometry, and provides interesting results when negated ...
Maren Morris is embracing a new era of independence, both in her professional and personal life.On Thursday, the Grammy-winning star drops an emotional cover of Billy Idol's "Dancing With Myself ...
Mutual information is a measure of the inherent dependence expressed in the joint distribution of and relative to the marginal distribution of and under the assumption of independence. Mutual information therefore measures dependence in the following sense: I ( X ; Y ) = 0 {\displaystyle \operatorname {I} (X;Y)=0} if and only if X ...