Re: bias and variance  definition of g bar
Examination of the two charts for hypothesis H1 in the book brings up an interesting point that may be helpful. The top chart shows various lines (g's) plotted on the graph. Each line is the result of taking two randomly selected points in the x axis, evaluating them to get the y value and then putting a line thru the two points. Those points will always be on the sinusoid but the lines (the g's) representing the associated function g(x) do not have to be. Notice the two outlying lines in the upper left corner of the top chart. They are totally off of the sinusoid but if g(x)  gbar(x) was to be evaluated in that corner then the value of the outlier g's at that x value (close to 1) must be included in the determination of gbar(x). The only way to get those values is to calculate the y value of each associated line at that point.
