Add a square*:-

For the housing data find a best fit of the form Price ≈ a + b Age + c Sqft + d Sqft². (Linear estimation entropy*) Suppose that an unknown parameter x has a normal (Gaussian) distribution with variance σ². Independent measurements of the form yi = x +εi are made, where each εi is Gaussian with mean zero and variance ω² and is independent of x.

The entropy of a multidimensional Gaussian variable is H = ½ log (2πe) n|Q| bits, where |Q| is the determinant of the covariance matrix corresponding to the variable. (See chapter 22 for the n = 1 case.)

(a) Find I(y₁, y₂, ... , y_n; x)

(b) Let

And find I( ; x), showing that for purposes of estimation it is only necessary to consider y rather than keeping track of the individual yi's.