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-   -   Confused on question 6. (http://book.caltech.edu/bookforum/showthread.php?t=1993)

ArikB 10-07-2012 12:09 PM

Confused on question 6.
 
Edit: Solved it, the story below is no longer relevant. :)

I'm confused about how one is supposed to calculate the score exactly. My biggest confusion seems to stem from the misunderstanding of what a 'point' is. Is a point one of the input vectors? so 101, 110 and 111 are 3 points?

So then g[a], returns 1 for all three points would mean that:

Code:

101 | 1
110 | 1
111 | 1

And g[b], returns all 0's:

Code:

101 | 0
110 | 0
111 | 0

And g[c], the xor function, would return:

Code:

101 | 0
110 | 0
111 | 1

and g(d), inverse of g(c), would return:

Code:

101 | 1
110 | 1
111 | 0

Or could it be that g[1] means that it will only return a 1 if all points are 1? So:

Code:

101 | 0
110 | 0
111 | 1

and g[b] would have a score of 0, because there are no 000 points.

I'm utterly confused by the question. :/

yaser 10-07-2012 02:02 PM

Re: Confused on question 6.
 
Quote:

Originally Posted by ArikB (Post 6143)
I'm confused about how one is supposed to calculate the score exactly. My biggest confusion seems to stem from the misunderstanding of what a 'point' is. Is a point one of the input vectors? so 101, 110 and 111 are 3 points?

A point is a data point, so these are 3 points. For each possible target function, there is a number of agreements (0,1,2 or3) with your hypothesis on these 3 points. We are keeping a tally of the number of agreements as we go through all possible target functions.

ArikB 10-07-2012 02:03 PM

Re: Confused on question 6.
 
Quote:

Originally Posted by yaser (Post 6145)
A point is a data point, so these are 3 points. For each possible target function, there is a number of agreements (0,1,2 or3) with your hypothesis on these 3 points. We are keeping a tally of the number of agreements as we go through all possible target functions.


Thank you for the response, I was approaching the question completely wrong but solved it in the meantime. :)

yaser 10-07-2012 02:09 PM

Re: Confused on question 6.
 
Quote:

Originally Posted by ArikB (Post 6146)
Thank you for the response, I was approaching the question completely wrong but solved it in the meantime. :)

You are welcome. Everyone is encouraged to ask questions, big or small.

noahdavis 10-08-2012 08:46 AM

Re: Confused on question 6.
 
Quote:

Originally Posted by yaser (Post 6145)
A point is a data point, so these are 3 points. For each possible target function, there is a number of agreements (0,1,2 or3) with your hypothesis on these 3 points. We are keeping a tally of the number of agreements as we go through all possible target functions.

Sorry I'm struggling a bit understanding the framework here. Maybe it's just terminology. What is the difference between a "possible target function" and a "hypothesis" ? I thought that they were equivalent, but it does not seem to be the case - a hypothesis must agree with a target function.

yaser 10-08-2012 10:31 AM

Re: Confused on question 6.
 
Quote:

Originally Posted by noahdavis (Post 6183)
Sorry I'm struggling a bit understanding the framework here. Maybe it's just terminology. What is the difference between a "possible target function" and a "hypothesis" ? I thought that they were equivalent, but it does not seem to be the case - a hypothesis must agree with a target function.

Possible target function is a notion introduced in this problem in order to make a point about learning. In general, there is one target function, albeit unknown. Here we spell out "unkown" by considering all the possibilities the target function can assume. We can afford to do that here because there is only a finite number of possibilities.

Hypotheses are the products of learning that try to approximate the target function. In this problem, we prescribe different learning scenarios that result in different hypotheses, then attempt to grade these hypotheses. We grade them according to how well each of them approximates the target function. The twist is that we consider all possible target functions and grade the hypothesis according to how well it approximates each of these possible targets.

noahdavis 10-08-2012 02:52 PM

Re: Confused on question 6.
 
Thank you - I understand now. For some reason it took me a leap to figure out how to build the "target function" such that it could be measured as stated in the problem. Originally, I had a list of 8 "functions" - but each function was just simply one of the 8 permutations where a permutation was an input point and a possible output.

apank 10-08-2012 06:29 PM

Re: Confused on question 6.
 
Hi,
What's a possible target function? Is that a combination of boolean operators? How do you come up with the formula 2^2^3 for total number of possibl target functions for 3 boollean inputs? Thank you.

yaser 10-08-2012 09:49 PM

Re: Confused on question 6.
 
Quote:

Originally Posted by apank (Post 6191)
Hi,
What's a possible target function? Is that a combination of boolean operators? How do you come up with the formula 2^2^3 for total number of possibl target functions for 3 boollean inputs? Thank you.

A possible target function is any function that could have generated the 5 data points in this problem, i.e., any function whose values on these five points all agree with the data.

There are 2^3=8 points in the input space here, which are all binary combinations of the 3 input variables from 000 to 111. For each of these points, a Boolean function may return 0 or 1; hence two possibilities. Therefore, for all 8 points, a Boolean function may return 2\times 2 \times \cdots \times 2 (8 times) possibilities, which gives us the number of different Boolean functions 2^{2^3}=2^8=256.

dobrokot 01-08-2013 01:26 PM

Re: Confused on question 6.
 
Quote:

score = (# of target functions agreeing with hypothesis on all 3 points)*...+...
"all 3 points" are points 101,110,111 outside D ? "2 points" are 2 points of given three?
So, y-values on points in D are not used in the answer?

Seems, number of matches do not affected which hypothesis I choose - any hypothesis produce same number of matches Binomial(3, #matches) on these 3 points. Seems too easy, like dangerous trap or puzzle with catchy answer - if number of matches is always the same, why to define some complicated functions of matches and give Y-values on other five points. Or I got something wrong :| May be matches outside these 3 points (matches inside D) should be counted too?


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