Minimum MSE estimation derivation

[EmmaSaunders1]

Hello,

Would anyone be-able to recommend a good, easy to read article which outlines MMSE and its derivation. Specifically I am having trouble finding this term


$$+ \int x'xP(x|y)dx-\left \| \int xP(x|y)dx \right \|^2$$

from

$$E({\left | \left | X-z \right | \right |}^2|Y=y) =\int (x-z)'(x-z)P(x|y)dx\\ =[z'-\int x'P(x|y)dx][z-\int xP(x|y)dx] + \int x'xP(x|y)dx-\left \| \int xP(x|y)dx \right \|^2$$

Thank you

 

[EmmaSaunders1]

Shouldnt the term just be zero - I can't understand it's presence - are there any conditions in which it is not zero??

 

[EmmaSaunders1]

For anyone who is interested - the last term

$$+ \int x'xP(x|y)dx-\left \| \int xP(x|y)dx \right \|^2$$

is necessary to account for the difference between E(x^2) and [E(x)]^2. When Z = E[x|Y=y] the term

$$E({\left | \left | X-z \right | \right |}^2|Y=y)$$

Is a minimum and reduces to

$$+ \int x'xP(x|y)dx-\left \| \int xP(x|y)dx \right \|^2 =$$

Then

$$E({\left | \left | X \right | \right |}^2|Y=y)-E(X|Y=y)^2\\ =E({\left | \left | X \right | \right |}^2|Y=y)-{\left | \left | \hat{X} \right | \right |}^2$$

which is the average mean square error