MIT6_079F09_lec07

MIT6_079F09_lec07 - Convex Optimization Boyd &...

Info iconThis preview shows pages 1–6. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

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

Unformatted text preview: Convex Optimization Boyd & Vandenberghe 7. Statistical estimation maximum likelihood estimation optimal detector design experiment design 71 Parametric distribution estimation distribution estimation problem: estimate probability density p ( y ) of a random variable from observed values parametric distribution estimation: choose from a family of densities p x ( y ) , indexed by a parameter x maximum likelihood estimation maximize (over x ) log p x ( y ) y is observed value l ( x ) = log p x ( y ) is called log-likelihood function can add constraints x C explicitly, or define p x ( y ) = 0 for x C a convex optimization problem if log p x ( y ) is concave in x for fixed y Statistical estimation 72 Linear measurements with IID noise linear measurement model y i = a i T x + v i , i = 1 , . . . , m x R n is vector of unknown parameters v i is IID measurement noise, with density p ( z ) R m producttext m T y i is measurement: y has density p x ( y ) = i =1 p ( y i a i x ) maximum likelihood estimate: any solution x of m maximize l ( x ) = i =1 log p ( y i a i T x ) ( y is observed value) Statistical estimation 73 summationdisplay summationdisplay braceleftbigg examples Gaussian noise N (0 , 2 ) : p ( z ) = (2 2 ) 1 / 2 e z 2 / (2 2 ) , m m 1 T l ( x ) = log(2 2 ) ( a i x y i ) 2 2 2 2 i =1 ML estimate is LS solution Laplacian noise: p ( z ) = (1 / (2 a )) e | z | /a , m 1 l ( x ) = m log(2 a ) | a i T x y i | a i =1 ML estimate is 1-norm solution uniform noise on [ a, a ] : m log(2 a ) | a i T x y i | a, i = 1 , . . . , m l ( x ) = otherwise ML estimate is any x with | a i T x y i | a Statistical estimation 74 summationdisplay summationdisplay...
View Full Document

This note was uploaded on 01/31/2012 for the course ECE 202 taught by Professor Boyde during the Fall '06 term at Goldsmiths.

Page1 / 16

MIT6_079F09_lec07 - Convex Optimization Boyd &...

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online