How is Chernoff bound calculated?

How is Chernoff bound calculated?

Using Chernoff bounds, find an upper bound on P(X≥αn), where p<α<1. Evaluate the bound for p=12 and α=34. For X∼Binomial(n,p), we have MX(s)=(pes+q)n, where q=1−p.

Is Chernoff always tighter than Markov?

Chernoff bounds are typically tighter than Markov’s inequality and Chebyshev bounds but they require stronger assumptions. First we will state our assumptions and definitions. Let X be a sum of n independent random variables {Xi}, with E[Xi] = pi.

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.

What are Chernoff bounds used for?

The Chernoff bound sometimes refers to the above inequality, which was first applied by Sergei Bernstein to prove the related Bernstein inequalities. It is also used to prove Hoeffding’s inequality, Bennett’s inequality, and McDiarmid’s inequality.

What is the MGF of exponential distribution?

Let X be a continuous random variable with an exponential distribution with parameter β for some β∈R>0. Then the moment generating function MX of X is given by: MX(t)=11−βt.

What is inequality in statistics?

Statistical Inequalities provide a means of bounding measures and quantities and are particularly useful in specifying bounds on quantities that may be difficult or intractable to compute. They also underpin a great deal of theory in Probability, Statistics, and Machine Learning.

What is moment bound?

A less well-known bound, also derivable from Markov’s. inequality, holds for positive t and is termed the moment. bound. This bound is obtained from (4) by taking h(x) = (x )n, for n a positive integer, and can be written as.

What is the moment generating function of normal distribution?

(8) The moment generating function corresponding to the normal probability density function N(x;µ, σ2) is the function Mx(t) = exp{µt + σ2t2/2}.

What are the applications of tail bounds?

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.