MIT15_097S12_lec15

2 12 i1 we see that the map estimate corresponds

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: riate normal prior (we will see later why this is a good idea) with mean 0. The covariance matrix will be diagonal, it is I (the 8 identity matrix) times σ 2 /C , which is the known, constant variance of each component of θ. So the prior is: θ ∼ N (0, Iσ 2 /C ) = �−1 1 1 � exp − θT Iσ 2 /C θ . 2 (2π )d/2 |σ 2 /C |1/2 Here we are saying that our prior belief is that the “slope” of the regression line is near 0. Now, ˆ θMAP ∈ arg max log p(y |x, θ) + log p(θ) θ m 1 m (yi − θT xi )2 + log p(θ) = arg max − 2 θ 2σ i=1 from (10). At this point you can really see how MAP is a regularized ML. Continuing, m 1 m d 1 σ 2 1 T T 2 (yi − θ xi ) − log 2π − log = arg max − 2 − θ (IC/σ 2 )θ θ 2 2 C 2 2σ i=1 m 1 m 1 (yi − θT xi )2 − 2 C θT θ = arg max − 2 θ 2σ i=1 2σ m m = arg min (yi − θT xi )2 + C IθI2 . 2 θ (12) i=1 We see that the MAP estimate corresponds exactly to R2 -regularized linear regression (ridge regression), and that the R2 reg...
View Full Document

This note was uploaded on 03/24/2014 for the course MIT 15.097 taught by Professor Cynthiarudin during the Spring '12 term at MIT.

Ask a homework question - tutors are online