LFD Book Forum Discussion of the VC proof
 User Name Remember Me? Password
 Register FAQ Calendar Mark Forums Read

 Thread Tools Display Modes
#21
10-26-2016, 02:28 PM
 CountVonCount Member Join Date: Oct 2016 Posts: 17
Re: Discussion of the VC proof

I have also another question on the same page (190):
At the end of the page there is the formula:

$\sum_S&space;\mathbb{P}[S]\times\mathbb{P}[sup_{h\in&space;H}\vert&space;E_{in}(h)&space;-&space;{E_{in}}'(h))&space;\vert&space;>&space;\frac{\varepsilon&space;}{2}]&space;\leq&space;sup_S&space;\mathbb{P}[sup_{h\in&space;H}\vert&space;E_{in}(h)&space;-&space;{E_{in}}'(h))&space;\vert&space;>&space;\frac{\varepsilon&space;}{2}]&space;\vert&space;S&space;]$

I don't understand why the RHS is greater or equal to the LHS. The only legitimation I see for this is, that the distribution of P[S] is uniform, but this has not been stated in the text.
Or do I oversee here anything and this is also valid for all kinds of distribution?
#22
10-26-2016, 02:32 PM
 CountVonCount Member Join Date: Oct 2016 Posts: 17
Re: Discussion of the VC proof

Quote:
 Originally Posted by magdon Suppose Then, . In which case and the bound in Theorem A.1 is trivial.
Hello,

thanks for the answer. I understand this argument, however this holds also for

or for

Thus the value 1/4 is somehow magic for me.

Edit: I think you choose 1/4 because it is so easy to see, that the RHS of Theorem A.1 gets 1. Nevertheless with a different value you would get a different outcome of the final formula.
#23
10-28-2016, 12:52 AM
 CountVonCount Member Join Date: Oct 2016 Posts: 17
Re: Discussion of the VC proof

Quote:
 Originally Posted by CountVonCount ... I don't understand why the RHS is greater or equal to the LHS. The only legitimation I see for this is, that the distribution of P[S] is uniform, but this has not been stated in the text. Or do I oversee here anything and this is also valid for all kinds of distribution?
I got it by myself.
When we have a uniform distribution of P[S] the outcome of the product-sum

is simply the average of all P[A|S], since

And an average is of course less than or equal to the maximum of P[A|S].
If P[S] is not distributed uniformly we still have an average, but a weighted average. But also here the result is always less than or equal to the maximum. Because you cannot find weighting factors that are in sum 1 but will lead to a higher result as the maximum P[A|S].
#24
11-09-2016, 05:21 AM
 magdon RPI Join Date: Aug 2009 Location: Troy, NY, USA. Posts: 594
Re: Discussion of the VC proof

Correct.

Quote:
 Originally Posted by CountVonCount I got it by myself. When we have a uniform distribution of P[S] the outcome of the product-sum is simply the average of all P[A|S], since And an average is of course less than or equal to the maximum of P[A|S]. If P[S] is not distributed uniformly we still have an average, but a weighted average. But also here the result is always less than or equal to the maximum. Because you cannot find weighting factors that are in sum 1 but will lead to a higher result as the maximum P[A|S].
__________________
Have faith in probability
#25
11-09-2016, 05:23 AM
 magdon RPI Join Date: Aug 2009 Location: Troy, NY, USA. Posts: 594
Re: Discussion of the VC proof

You are correct. We could have made other assumptions.

But there is a special reason why we DO want 1. Because we are bounding the probability, and so there is nothing to prove if we claim that a probability is less equal to 1. So, whenever the RHS (i.e. the bound) evaluates to 1 or bigger, there is nothing to prove. So we only need to consider the case when the bound evaluates to less than 1.

Quote:
 Originally Posted by CountVonCount Hello, thanks for the answer. I understand this argument, however this holds also for or for Thus the value 1/4 is somehow magic for me. Edit: I think you choose 1/4 because it is so easy to see, that the RHS of Theorem A.1 gets 1. Nevertheless with a different value you would get a different outcome of the final formula.
__________________
Have faith in probability
#26
11-11-2016, 02:52 AM
 CountVonCount Member Join Date: Oct 2016 Posts: 17
Re: Discussion of the VC proof

Thank you very much for your answers. This helps me a lot to understand the intention of the single steps of this prove.
#27
06-14-2017, 11:38 AM
 eh3an2010 Junior Member Join Date: Jun 2017 Posts: 1
Re: Discussion of the VC proof

I got it by myself.
When we have a uniform distribution of P[S] the outcome of the product-sum
__________________
[RIGHT][SIZE="2"]myblog: [url=https://sitedp.com]طراحی سایت[/url]
[url=https://sitedp.com]طراحی وب سایت[/url][/SIZE][/RIGHT]
#28
07-29-2017, 12:13 AM
 alireza-res Junior Member Join Date: Jul 2017 Posts: 1
Re: Discussion of the VC proof

i agree with you , thank you very much
__________________
my weblog: [URL="http://respinatech.com/"]دیجیتال مارکتینگ[/URL] - [URL="http://www.drghanei.ir/laser-hair/"]لیزر موهای زائد[/URL]
#29
08-28-2017, 09:13 AM
 sarafox Junior Member Join Date: Aug 2017 Posts: 2
Re: Discussion of the VC proof

i had same question but thanks to your website my question just solved. thanks
__________________
https://seofox.ir
#30
12-07-2017, 02:28 AM
 babak Junior Member Join Date: Dec 2017 Posts: 3
Re: Discussion of the VC proof

I just want to say thank you, my problem is solved
__________________
[URL="https://rayatarh.com/"]دیجیتال مارکتینگ[/URL] [URL="http://www.otag.ir/"]دکوراسیون منزل[/URL] [URL="https://rayatarh.com/google-adwords/"]تبلیغ در گوگل[/URL]

 Thread Tools Display Modes Linear Mode

 Posting Rules You may not post new threads You may not post replies You may not post attachments You may not edit your posts BB code is On Smilies are On [IMG] code is On HTML code is Off Forum Rules
 Forum Jump User Control Panel Private Messages Subscriptions Who's Online Search Forums Forums Home General     General Discussion of Machine Learning     Free Additional Material         Dynamic e-Chapters         Dynamic e-Appendices Course Discussions     Online LFD course         General comments on the course         Homework 1         Homework 2         Homework 3         Homework 4         Homework 5         Homework 6         Homework 7         Homework 8         The Final         Create New Homework Problems Book Feedback - Learning From Data     General comments on the book     Chapter 1 - The Learning Problem     Chapter 2 - Training versus Testing     Chapter 3 - The Linear Model     Chapter 4 - Overfitting     Chapter 5 - Three Learning Principles     e-Chapter 6 - Similarity Based Methods     e-Chapter 7 - Neural Networks     e-Chapter 8 - Support Vector Machines     e-Chapter 9 - Learning Aides     Appendix and Notation     e-Appendices

All times are GMT -7. The time now is 09:08 PM.

 Contact Us - LFD Book - Top

Powered by vBulletin® Version 3.8.3
Copyright ©2000 - 2018, Jelsoft Enterprises Ltd.
The contents of this forum are to be used ONLY by readers of the Learning From Data book by Yaser S. Abu-Mostafa, Malik Magdon-Ismail, and Hsuan-Tien Lin, and participants in the Learning From Data MOOC by Yaser S. Abu-Mostafa. No part of these contents is to be communicated or made accessible to ANY other person or entity.