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MIT6_079F09_lec07

MIT6_079F09_lec07 - Convex Optimization Boyd Vandenberghe 7...

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Convex Optimization Boyd & Vandenberghe 7. Statistical estimation maximum likelihood estimation optimal detector design experiment design 7–1
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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 7–2
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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 7–3
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summationdisplay summationdisplay braceleftbigg examples Gaussian noise N (0 , σ 2 ) : p ( z ) = (2 πσ 2 ) 1 / 2 e z 2 / (2 σ 2 ) , m m 1 T l ( x ) = log(2 πσ 2 ) ( a i x y i ) 2 2 2 σ 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 7–4
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summationdisplay summationdisplay Logistic regression random variable y ∈ { 0 , 1 }
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