Table of Contents
What does the Chernoff bound do?
In probability theory, the Chernoff bound gives exponentially decreasing bounds on tail distributions of sums of independent random variables.
How is Chernoff bound calculated?
Thus, the Chernoff bound for P(X≥a) can be written as P(X≥αn)≤mins>0e−saMX(s) =mins>0e−sa(pes+q)n.
How tight is Chernoff bound?
How tight is Chernoff bound? small deviations. It is shown here that the sum deviates from its mean µ with standard deviation of at least √ µ.
Is Chernoff always tighter than Markov?
Chernoff bounds are typically tighter than Markov’s inequality and Chebyshev bounds but they require stronger assumptions.
Which are the tail bound?
What is a Tail Bound? The tails of a random variable X are those parts of the probability mass function far from the mean [1]. Sometimes we want to create tail bounds (or tail inequalities) on the PMF, or bound the probability that the random variable deviates a long way from the mean.
What is tail bound?
In probabilistic analysis, we often need to bound the probability that a. random variable deviates far from its mean. There are various formulas. for this purpose. These are called tail bounds.
Is Chebyshev stronger than Markov?
The Markov inequality is one of the major tools for establishing probability bounds on the runtime of algorithms. If as well as the mean, the variance is known, a bound due to Chebyshev can be used, which is much stronger than that of Markov.
What is application of tail bound in big data?
In Machine Learning tail bounds help quantifying the extraction of information from large data sets by estimating the probability for a learning algorithm to be approximately correct. Typical bounds quantify the deviation of sample means from the exact expectation.
Is chi squared sub exponential?
It may be noted that a chisquare random variable is a special case of a sub-exponential random variable. There are several equivalent definitions of sub-exponential random variables. The one we find convenient is given as follows (see [5], p 26).
What is the application of tail bounds in big data?
What are concentration bounds?
Concentration bounds are inequalities that bound probabilities of deviations by a random variable from some value, often its mean. Informally, they show the probability that a random variable deviates from its expectation is small.
What is the first moment of a binomial distribution?
1 Answer. The expected value is sometimes known as the first moment of a probability distribution. The expected value is comparable to the mean of a population or sample.
Is Markov inequality tight?
We’ve found a probability distribution for X and a positive real number k such that the bound given by Markov’s inequality is exact; we say that Markov’s inequality is tight in the sense that in general, no better bound (using only E(X)) is possible. Markov’s inequality provides a terrible bound.
What are tail bounds?
What is Chernoff bound in statistics?
Chernoff bound. In probability theory, the Chernoff bound, named after Herman Chernoff but due to Herman Rubin, gives exponentially decreasing bounds on tail distributions of sums of independent random variables.
Why is it called Chernoff’s law?
Despite being named after Herman Chernoff, the author of the paper it first appeared in, the result is due to Herman Rubin. It is a sharper bound than the known first- or second-moment-based tail bounds such as Markov’s inequality or Chebyshev’s inequality, which only yield power-law bounds on tail decay.
Is Chernoff bound the same as Markov inequality?
However, the Chernoff bound requires that the variates be independent – a condition that neither Markov’s inequality nor Chebyshev’s inequality require, although Chebyshev’s inequality does require the variates to be pairwise independent. It is related to the (historically prior) Bernstein inequalities and to Hoeffding’s inequality .
What is Chernoff’s inequality used for?
It is also used to prove Hoeffding’s inequality, Bennett’s inequality, and McDiarmid’s inequality . This inequality can be applied generally to various classes of distributions, including sub-gaussian distributions, sub- gamma distributions, and sums of independent random variables.. Chernoff bounds commonly refer to the case where