Question 1

[Humble]

What does the outside expected value E[E(Wlin)] value mean in words.

[yaser]

Originally Posted by Humble

What does the outside expected value E[E(Wlin)] value mean in words.

First, just to make sure, the inside 'E' is not an expectation, but the value of the in-sample error that corresponds to the weight vector . The (outside) expected value is with respect to the training data set, and it means the average value (of the in-sample error) as you train with different data sets.

[hsolo]

Originally Posted by yaser

First, just to make sure, the inside 'E' is not an expectation, but the value of the in-sample error that corresponds to the weight vector . The (outside) expected value is with respect to the training data set, and it means the average value (of the in-sample error) as you train with different data sets.

Training data has d dimensions in the x's. If one ignored some of the dimensions and did linear regression with reduced number d' of dimensions one would have larger in-sample errors presumably, compared to considering all d dimensions?

Why then is the expected in-sample error averaged over all data sets increasing with the number of dimensions?

[yaser]

Originally Posted by hsolo

Training data has d dimensions in the x's. If one ignored some of the dimensions and did linear regression with reduced number d' of dimensions one would have larger in-sample errors presumably, compared to considering all d dimensions?

Why then is the expected in-sample error averaged over all data sets increasing with the number of dimensions?

To answer the first question, if you choose to omit some of the input variables, you will indeed get a larger (at least not smaller) in-sample error. Not sure I understand the second question, but having different training sets does not change the number of input variables. It is a hypothetical situation where you assume the availability of different data sets on the same variables.

[hsolo]

Originally Posted by yaser

To answer the first question, if you choose to omit some of the input variables, you will indeed get a larger (at least not smaller) in-sample error. Not sure I understand the second question, but having different training sets does not change the number of input variables. It is a hypothetical situation where you assume the availability of different data sets on the same variables.

My bad for the second question -- I had a typo in my handwritten expression for the expectation. The correct expression does have expected in-sample error decreasing as d is increasing.