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In probability theory, Boole's inequality, also known as the union bound, says that for any finite or countable set of events, the probability that at least one of the events happens is no greater than the sum of the probabilities of the individual events.
In fact, the union bound states that the probability of union of some events is smaller than the first term in the inclusion-exclusion formula. We can in fact extend the union bound to obtain lower and upper bounds on the probability of union of events.
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- 6.3.1 The Union Bound
- 6.3.3 Jensen's Inequality
- 6.3.4 Hoe ding's Inequality
In this section, we will talk about a potpourri of remaining concentration bounds. More speci cally, the union bound, Jensen's inequality for convex functions, and Hoe ding's inequality.
Suppose there are many bad events B1; : : : ; Bn, and we don't want any of them to happen. They may or may not be independent. Can we bound the probability that any (at least one) bad event occurs? The intuition for the union bound is fairly simple. Suppose we have two events A and B. Then P (A [ B) P (A) + P (B) since the event space of A and B ma...
Now after learning about convex sets and functions, we can learn Jensen's inequality, which relates E [g(X)] and g(E [X]) for convex functions. Remember we said many times that these two quantities were never equal (use LOTUS to compute E [g(X)])!
One nal inequality that is commonly used is called Hoe ding's inequality. We'll state it without proof since it is quite complicated. The proof uses Jensen's inequality and ideas from the proof of the Cherno bound (MGFs)!
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The union bound is a fundamental principle in probability theory that provides an upper limit on the probability of the union of multiple events. It states that the probability of at least one of several events occurring is less than or equal to the sum of their individual probabilities.
Aug 29, 2021 · the result Finite Union of Sets in Subadditive Function which gives: $\ds \map f {\bigcup_{i \mathop = 1}^n A_i} \le \sum_{i \mathop = 1}^n \map f {A_i}$ for a subadditive function $f$. $\blacksquare$ Also known as. This inequality is also known as union bound. Source of Name. This entry was named for George Boole. Sources
Boole's inequality (or the union bound ) states that for any at most countable collection of events, the probability that at least one of the events happens is no greater than the sum of the probabilities of the events in the collection.
Concentration Inequalities and Union Bound. This note introduces the basics of concentration inequalities and examples of its applications (often with union bound), which will be useful for the rest of this course. Theorem 1.
