Chernoff Bound for Binomial Distribution

In summary, the Chernoff Bound for Binomial Distribution is a mathematical tool used to estimate the probability of deviation from expected value in a large number of independent and identically distributed random variables. It is calculated using the Chernoff-Hoeffding inequality and takes into account the number of variables, probability of success, and desired deviation. It is significant in fields such as statistics and machine learning and is considered one of the strongest bounds due to its versatility and computational efficiency. It can also be applied to other types of distributions with additional calculations and adjustments.
  • #1
EngWiPy
1,368
61
Hello,

I've read in a paper that the following binomial distribution

[tex]\sum_{k=floor(N/2)+1}^N{N\choose k}\varepsilon^k(1-\varepsilon)^{N-k}[/tex]

can be upper bounded using Chernoff bound by

[tex]e^{ floor(N/2)}\,\Phi(s_0)[/tex]

where

[tex]\Phi(s)=\left(1-\varepsilon(1-e^s)\right)^N[/tex]

and

[tex]floor(N/2)\,\Phi(s_0)=\frac{\partial}{\partial s}\left. \Phi(s)\right|_{s=s_0}[/tex]

Could anyone explain to me how?

Thanks
 
Physics news on Phys.org

Related to Chernoff Bound for Binomial Distribution

1. What is the Chernoff Bound for Binomial Distribution?

The Chernoff Bound for Binomial Distribution is a mathematical tool used to estimate the probability that the sum of a large number of independent and identically distributed random variables will deviate from its expected value by a certain amount.

2. How is the Chernoff Bound for Binomial Distribution calculated?

The Chernoff Bound for Binomial Distribution is calculated using the Chernoff-Hoeffding inequality, which is a generalization of the Chernoff Bound for any distribution. It takes into account the number of random variables, the probability of success for each variable, and the desired deviation from the expected value.

3. What is the significance of the Chernoff Bound for Binomial Distribution?

The Chernoff Bound for Binomial Distribution is significant because it allows scientists to make accurate estimates and predictions about the behavior of large numbers of independent and identically distributed random variables. It is commonly used in fields such as statistics, probability, and machine learning.

4. How does the Chernoff Bound for Binomial Distribution compare to other bounds?

The Chernoff Bound for Binomial Distribution is considered one of the strongest bounds due to its ability to handle a wide range of distributions and its tightness in many cases. It is also computationally efficient, making it a popular choice for practical applications.

5. Can the Chernoff Bound for Binomial Distribution be applied to other types of distributions?

Yes, the Chernoff Bound can be applied to any distribution, not just binomial. However, it may require additional calculations and adjustments depending on the specific distribution being used. Other common applications of the Chernoff Bound include normal, Poisson, and exponential distributions.

Similar threads

Chernoff Bounds using importance sampling identity
    • Set Theory, Logic, Probability, Statistics
Replies
0
Views
1K
MHB Minimum Degree of a Random Graph (Probabilistic Method)
    • Set Theory, Logic, Probability, Statistics
Replies
1
Views
1K
Lower Bound on Binomial Distribution
    • Set Theory, Logic, Probability, Statistics
Replies
15
Views
2K
MHB Forward Unit Push Operator Equation
    • Set Theory, Logic, Probability, Statistics
Replies
4
Views
784
A How to derive the sampling distribution of some statistics
    • Set Theory, Logic, Probability, Statistics
Replies
3
Views
895
A A different discrete normal distribution
    • Set Theory, Logic, Probability, Statistics
Replies
2
Views
1K
I Casella Berger: Why is distribution of F-statistic in ANOVA not T^2
    • Set Theory, Logic, Probability, Statistics
Replies
1
Views
1K
MHB Binomial Distribution: Likelihood Ratio Test for Equality of Several Proportions
    • Set Theory, Logic, Probability, Statistics
Replies
1
Views
2K
A The distribution that has a certain distribution as its limit case
    • Set Theory, Logic, Probability, Statistics
Replies
7
Views
1K
I Probability of White Ball in Box of 120 Balls: Solved!
    • Set Theory, Logic, Probability, Statistics
Replies
3
Views
2K

Hot Threads

Recent Insights

Back
Top