Bias-Variance Analysis

[Andrew87]

Hello,

I'm getting confused about . Why is it the best approximation of the target function we could obtain in the unreal case of infinite training sets ?

Thank you in advance,
Andrea

[yaser]

Originally Posted by Andrew87

Hello,

I'm getting confused about . Why is it the best approximation of the target function we could obtain in the unreal case of infinite training sets ?

Thank you in advance,
Andrea

It is not necessarily the best approximation of the target function, but it is often close. If we have one, infinite-size training set, and we have infinite computational power that goes with it, we can arrive at the best approximation. In the bias-variance analysis, we are given an infinite number of finite training sets, and we are restricted to using one of these finite training sets at a time, then averaging the resulting hypotheses. This restriction can take us away from the absolute optimal, but usually not by much.

[Andrew87]

Thank you very much for your answer Prof. Yaser. It clarified my doubt.

My kind regards,
Andrea

[sayan751]

Hi,

I have a doubt regarding g bar.

I tried to calculate the bias for the second learner, i.e. h(x) = ax + b. So this is how did it:

• Generated around 1000 data points (x ranging from -1 to 1)
• Then picked up two sample data points at random
• Solved for a and b using matrix
• Repeated this process for around 3000 times and
• Lastly took mean for a and mean for b, which formed the g2 bar
• Used this g2 bar for calculating the respective bias, which also matched with the given value of bias

Now I have two questions:
1. Please let me know whether I am proceeding in the right direction or not.
2. When I am trying to repeat this process with a polynomial model instead of linear model, my calculated bias for the polynomial model varies in great margin, even if the sample data points doesn't change. For polynomial as well, I took the mean of the coefficients, but still my answer (both g bar and bias) varies greatly with each run. What I am missing here?

[yaser]

Originally Posted by sayan751

1. Please let me know whether I am proceeding in the right direction or not.
2. When I am trying to repeat this process with a polynomial model instead of linear model, my calculated bias for the polynomial model varies in great margin, even if the sample data points doesn't change. For polynomial as well, I took the mean of the coefficients, but still my answer (both g bar and bias) varies greatly with each run. What I am missing here?

1. Your approach is correct. While sampling from a fixed 1000-point set is not the same as sampling from the whole domain, it should be close enough.

2. Not sure if this is the reason, but if you are still using a 2-point training set, a polynomial model will have too many parameters, leading to non-unique solutions that could vary wildly.

[sayan751]

Thank You Prof. Yaser for your reply.

I am using a 10 point dataset for the polynomial model. However, the problem I am referring to defines y = f(x) + noise = x + noise.

Previously by mistake I was assuming f(x) as y rather than only x. Later I noticed that all the calculation of bias and variance concentrate purely on f(x). Hence later I ignored the noise and now I am getting stable bias and variance for polynomial model for each run.

[Jackwsimpson]

I am confused in trying to get from the first line to the second line for the first set of equations on page 63: ... ED[Ex[(g... on the first line to ...Ex[ED[( on the second line.

I sort of see the first line: expected value with respect to data set x (a subset of D I assume) is averaged over all possible data set x's in D. On the second line we have what might be the average of the argument over all of D inside the outer brackets. I don't know how to interpret Ex outside the outer brackets.

In short, I certainly don't understand what exactly is meant by the 2nd line, and I may well not understand the first line. Any further explanation possible?