map - Boston University Department of Electrical and...

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Unformatted text preview: Boston University Department of Electrical and Computer Engineering EC505 STOCHASTIC PROCESSES Notes on MAP Estimation 1 Bayes Maximum Aposteriori Estimation In this set of notes we will discuss Bayesian Maximum Aposteriori (MAP) estimation. In class we learned that the MAP estimator corresponds to the Bayesian estimator for the cost function: J MAP ( x , b x ( y )) = 1 x 6 = b x ( y ) x = b x ( y ) (1) Recall that the Bayesian estimator minimizes the expected value of the cost, thus: b x MAP = arg min x E J MAP ( x , b x ( y )) = arg min x E J MAP ( x , b x ( y )) y (2) In class we showed that this estimate maximizes the posterior density: b x MAP = arg max x p X | Y ( x | y ) = arg max x p Y | X ( y | x ) p X ( x ) (3) where p Y | X ( y | x ) is known as the data likelihood and p X ( x ) is known as the prior. MAP Estimation for Gaussian Problems with Linear Observations Let us now focus on MAP estimation for problems with linear observations, Gaussian densities, and vector state. In particular, assume we have the following general problem, wherein our noisy vector observation y is linearly related to our quantity of interest x , which itself is Gaussian: y = Cx + w , w ∼ N (0 , R ) (4) x ∼ N (0 , Q ) (5) where y = [ y 1 , ··· , y N ] T , x = [ x 1 , ··· , x N ] T , w = [ w 1 , ··· , w N ] T , R is the covariance matrix of the observation noise, w is independent of x , and Q is the covariance matrix of x . In class we saw that the MAP estimator for the vector Gaussian case was the same as the Bayes and LLSE estimators. The form of the expressions we found there for the estimate, while correct, will prove inconvenient for our goals, and thus we will derive an alternative expression for the MAP estimate based directly on the MAP equation. Without loss of generality we will focus on the zero mean case. To this enddirectly on the MAP equation....
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This note was uploaded on 09/29/2009 for the course EC 505 taught by Professor Karl during the Fall '04 term at BU.

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map - Boston University Department of Electrical and...

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