Question

1 ( We have seen that, as the size of a data set increases, the uncertainty associated with the posterior distribution over model parameters decreases. Make use of the matrix identity (Appendix C) (M- v) (vTM-) 1vTM-v V (M vT)M м-1 (3.110 to show that the uncertainty o(x) associated with the lnear regression function given by (3.59) satis fies oN+(x)o(x) (3.111)

Answer 1

Answer:

+ φ(x) TSN (x) B B is the precision (inverse variance) of noise. The first term represents the noise on the data; the second term reflects the uncertainty associated with the parameters.

As SN 1 S + βΦΦ Ν S +βΣφχ) φ(χ,) 1 n-1 So, N+1 ΣφΧ)φ(xη) Τ. 1 S 1 N+1 Ss +β n-1 SN+ βφ(ΧN+1 ) φ (χn+1) , (x ) as we can express σ 'Ν+1 1 + φ(x)'Sh+1 (x) β σλ. (x) σλ(x)+ φ(x) " (SN+1-Sh)φ(x)

Applying the matrix identity to Eqn (1), by setting M SN and ν=Vφ(xN+1): BSNP (XN+1)(XN+1)^S» 1 B(XN+1)TSN(XN+1) SN+1 SN Substituting σn-1(x ) σ(x) - βφ(x) Snφ (Χν.+1 ) φ(χν+1 ) ' Sφ(x) 1+ Bp(XN+1)TSND(XN+1) Since SN is positive definite and SNø(xN+1)¢(XN+1)*SN is positive semidefinite, the second term above are nonnegative. Hence ON+1(x)o(x)