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-   -   Hoeffding Inequality Definition (Chapter 1, Eqn. 1.4) (http://book.caltech.edu/bookforum/showthread.php?t=4781)

SpencerNorris 09-14-2017 09:55 AM

Hoeffding Inequality Definition (Chapter 1, Eqn. 1.4)
 
I'm currently working on Exercise 1.9 and had a question about the Hoeffding inequality. I'm looking at the inner inequality of the left hand term, but I've seen the inequality presented in different ways in different locations. The textbook (p. 19) says |v - mu| strictly greater than epsilon, but Wikipedia claims |v - mu| greater than or equal to epsilon; same for UMich's Stat Learning Theory course: http://bit.ly/2x1RRkD .

I'm basically wondering if there's some sort of subtlety that I'm missing or if it was a mistake in the textbook. Thanks!

Spencer Norris

mauriciogruppi 09-15-2017 10:55 PM

Re: Hoeffding Inequality Definition (Chapter 1, Eqn. 1.4)
 
Quote:

Originally Posted by SpencerNorris (Post 12754)
I'm currently working on Exercise 1.9 and had a question about the Hoeffding inequality. I'm looking at the inner inequality of the left hand term, but I've seen the inequality presented in different ways in different locations. The textbook (p. 19) says |v - mu| strictly greater than epsilon, but Wikipedia claims |v - mu| greater than or equal to epsilon; same for UMich's Stat Learning Theory course: http://bit.ly/2x1RRkD .

I'm basically wondering if there's some sort of subtlety that I'm missing or if it was a mistake in the textbook. Thanks!

Spencer Norris

In cases like this, you can use greater than or equal to epsilon.

This has been explained by professor Malik here:
http://book.caltech.edu/bookforum/showthread.php?t=4412

magdon 09-17-2017 08:45 AM

Re: Hoeffding Inequality Definition (Chapter 1, Eqn. 1.4)
 
Technically, one can prove the Hoeffding inequality with

|\nu-\mu|\ge \epsilon

The one in the book with |\nu-\mu|> \epsilon is also true because the "BAD" event |\nu-\mu|> \epsilon is a "smaller" event than the "BAD" event |\nu-\mu|\ge \epsilon.

We wanted to define "GOOD" as |\nu-\mu|\le \epsilon, which means we should define the "BAD" event where you got fooled as |\nu-\mu|> \epsilon. This minor technicality has little or no practical significance.


Quote:

Originally Posted by SpencerNorris (Post 12754)
I'm currently working on Exercise 1.9 and had a question about the Hoeffding inequality. I'm looking at the inner inequality of the left hand term, but I've seen the inequality presented in different ways in different locations. The textbook (p. 19) says |v - mu| strictly greater than epsilon, but Wikipedia claims |v - mu| greater than or equal to epsilon; same for UMich's Stat Learning Theory course: http://bit.ly/2x1RRkD .

I'm basically wondering if there's some sort of subtlety that I'm missing or if it was a mistake in the textbook. Thanks!

Spencer Norris



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