#1




Probability calculation in Q8 and Q10
I correctly approximated the P(f(x) != g(x)) by generating a lot of sample points. However, how do you calculate the probability exactly? Thank you in advance for your help!

#2




Re: Probability calculation in Q8 and Q10
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#3




Re: Probability calculation in Q8 and Q10
I would be curious on how to calculate  or estimate  the average number of iterations for convergence  as well as the probability  perhaps these are really the same question in disguise. It is a very simple procedure, but there is enough random elements to it so that it leaves the realm of what I am familiar with calculating:
As I understand it ... 1) choose N random points in the unit square 2) choose a random line This gives the target 3) start with w(0) = [0;0] 4) run algorithm where in computing w(n+1) a random selection from the misclassified points is used. (perhaps the random choice here is notrelevant) A pointer to where a similar calculation is made would be fine. Thanks! 
#4




Re: Probability calculation in Q8 and Q10
Calculating these quantities analytically is not tractable. Estimating them using Monte Carlo methods, i.e., by running many random instances of the problem and averaging, is what we are after here. As you point out, there are many sources of randomness and some will result in significant variation. However, repeating the experiment a large number of times will overcome that variance. The numbers given for this problem were chosen to achieve that.
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#5




Re: Probability calculation in Q8 and Q10
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It looks a little similar to the relaxation algorithms for solving partial differential equations. In that case, the theory is well studied. Even without knowing that, you could find the scaling law. How the number of iterations changes, on the average, with the number of points. But once you have the MC program, it is easy enough to change the number of points. 
#6




Re: Probability calculation in Q8 and Q10
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#7




Re: Probability calculation in Q8 and Q10
Your value of g, assuming it doesn't exactly match f, will be a line with a slope and intercept close, though not exactly that of f. Imagine then, two intersecting lines. The area in between the lines correspond to values in X that g misclassifies. The ratio between that area and that bounded by the region [1,1]x[1,1] (area 4) will be the probability of error. Alas, calculating the area between the lines is ugly. The problem would have been much simpler if the domain for X had been points inside the circle at (0,0) with radius 1.

#8




Re: Probability calculation in Q8 and Q10
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If the input space is 2D, I can see that there are two triangles formed by f(x) and g(x), and the total area of the two triangles include all the errors. So I thought the P(f(x) != g(x)) = area of error / total area of input space...no? Thanks in advance. 
#9




Re: Probability calculation in Q8 and Q10
The total area of the input space is 4 and that corresponds to total probability 1, so one needs to normalize areas by 4 in order to get the corresponding probabilities.
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#10




Re: Probability calculation in Q8 and Q10
Argh, I see.
Missed the boundaries stated in the problem. My bad. Thanks. 
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