What is the definition of expectation with respect to data set
 

In probability class, we take expectation with respect to random variables with a certain probability distribution.

Suppose that X ~ N(mu,sigma^2) is a Gaussian random variable.

Then E[X] = mu, the mean of the Gaussian random variable, which can be show by performing expectation integral of the Gaussian distribution.

In the book, D is a set of pair of values {(x_n,y_n)}, not a random variable. Even if it is a random variable, its distribution is unknown.

Then what does does the symbol E_D or E_D_val mean?

What is the random variable here? Is it g, g^-, x_n, y_n, or (x_n,y_n) or e(g^-(x_n),y_n)?

And these random variables to generated according to what probability distribution?

This is the only part of the book that is confusing for me. Please clarify what it means to take expectation with respect to a set.

Re: What is the definition of expectation with respect to data set
 

A random variable does not need to be a scalar. It can be a vector, a matrix, etc. as long as it comes from a (measurable) set of outcomes.

In the case you are asking, the random variable is the dataset itself, coming from the family of all possible datasets. Its distribution is derived from the distribution that generates each .

Hope that this helps.

Re: What is the definition of expectation with respect to data set
 

Thank you for the answer.

However, there is still a distinction between a vector, and a set of vectors.

In other words, {1} is not the same as 1.

However, in multiple locations in the textbook, the dataset is represented as a set. For example, at the bottom of page 8.

My question is, is the dataset a vector consisting of pairs, or a set of pairs?

In other words, the distinction between [(1,2), (3,4)]^T versus {(1,2), (3,4)}. I don't really understand the notation D = (x_1, y_1), (x_2, y_2)...(x_N,y_N) because as it is written it is neither a vector nor a set.

I apologize in advance if I am being too rigorous and nitpicky

Re: What is the definition of expectation with respect to data set
 

The data set is a set of pairs in this book. While the set is by definition not ordered, the data set is more often "ordered by index"---and thus conveniently represented by a (pair of) matrix (like what we did in Chapter 3).

Hope this helps.