Question

Y = f(X) +€, e~ N(0,0%) Let $ ({X})$ be the estimate (or predicted model) of $ f({X})$. The mean squared error (MSE) at a data point x is E[(Y – f (x))] = (E[f (x)] – f(x))? + E[(f (x) – E[ f (x)])?] + o2 = Bias? + Variance + Irreducible Error

Answer 1

solution Given that let f^(x) be the estimate of f(x)" y = f(x) + €, ExN60102) He9te, - $ (lxb) $ be the estimate of $ $ ({x})$. Now We have to show that, the mean squared coooo (Mst) at a data point x les E[cy – f(x)) = (EPU) – f%)? + Elfex) – 4f087)_04 • Bias + vasånce + Tooeducible fooor 6(E)-0. Var (@)= 6? = E LEY) Mean squared error = Exyn f Cou? poodicted volus on Parlulas date set. . - € { l-8) + { _ f Cove) - E [CH6w) – cm) *+&+ 2 (f() – ca) 2)] - E[lf (x) – FlBv2) + E [ɛ] +2E [Cf (a) – € (278) ECP Ga) – Â (1339) +*+REC+ (2) _FGx1ECE).

= E ((f(x) - Î (2) 2) to? & NOW, we have to show that Elf (8) – $ Cay] E [ Cf (a) – E (fca]+ E [ẾCa] = f(x)e] Adding and qubtrabe. E ſf ()] three four • ELE[f'(x) = f (0) + E [fica) –EC€ (a)"} - 2E [Cf(4) -ELf(a) [f'ca) – E Cf(a))] - (E[fca] – f(a)?+ E [Cfca) – € [fca)] I varience. - 284ftas - Effa)). Ethos-fcẬCa) bias = bias [f cang? + vay [fo Average mean squared soov = bias [fcas} + vad flay) ++

Three four, We Get, Average mean squared eomos = biasff cay" i vanlfca] +02 - *****