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Unformatted text preview: Detection and Estimation (Spring, 2009) Bayesian Estimators NCTU EE P.1 Bayesian Estimators 9 Now we assume we have some prior knowledge about θ . To incorporate it, we assume θ is random variable with a given pdf. 9 Bayesian estimation is useful in situations where an MVU estimator cannot be found. ~ Prior Knowledge and Estimation Example: DC Level in WGN, but –A ≤ A ≤ A instead of  ∞ ≤ A ≤ ∞ . MVU estimator: x A = ˆ Truncated sample mean estimator: A ( is better than A ˆ in terms of MSE. We reduce the mean square error by allowing the estimator to be biased! Detection & Estimation (Spring, 2009) Bayesian Estimators NCTU EE P.2 Bayesian Approach: A is considered to be a random variable with a prior pdf. We attempt to estimate the realization of A. e.g., Assume A ~ U [A ,A ]. Classical: Bayesian: Remark: MSE depends on A. BMSE doesn’t. Since p( x ) ≥ 0 for all x , if the integral in brackets can be minimized for each x , then the Bayesian MSE will be minimized. Detection & Estimation (Spring, 2009) Bayesian Estimators NCTU EE P.3 Remarks: 9 The optimal estimator in terms of minimizing the Bayesian MSE is the mean of the posterior pdf p(A x ). 9 : the prior pdf of A 9 The estimator that minimizes the Bayesian MSE is termed the minimum mean square error (MMSE) estimator . Example: p(A) = U [A ,A ]. Assume w[n] is independent of A. where Detection & Estimation (Spring, 2009) Bayesian Estimators NCTU EE P.4 The MMSE estimator: Cannot be evaluated in closed form. Remarks: 9 A ˆ is biased towards zero unless N A 2 σ >> . In general, A ˆ is biased towards the prior mean. 9 As N increases, the MMSE estimator relies less and less on the prior knowledge and more on the data. 9 Before observation, we assume a prior pdf p( θ ). After observation, our state of knowledge about the parameter is summarized by the posterior pdf p( θ  x ). 9 The choice of a prior pdf is critical in Bayesian estimation. A wrong choice will result in a poor estimator. 9 An optimal estimator is defined to be the one that minimizes the MSE when averaged over all realizations of θ and x . MMSE estimator Detection & Estimation (Spring, 2009) Bayesian Estimators NCTU EE P.5 ~ Choosing a Prior PDF 9 Should be based on physical constraints of the problem. 9 Choose prior to allow easy integration. Example: DC Level in WGN – Gaussian Prior pdf Detection & Estimation (Spring, 2009) Bayesian Estimators NCTU EE P.6 Let The posterior pdf is also Gaussian, but with different mean and variance. where Remark: α is a weighting factor (0 < α < 1) that compromise between the prior knowledge and the data knowledge. Detection & Estimation (Spring, 2009) Bayesian Estimators NCTU EE P.7 The prior knowledge improves the estimation accuracy....
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This note was uploaded on 07/21/2009 for the course EE IEE5703 taught by Professor Shengjyhwang during the Spring '09 term at National Chiao Tung University.
 Spring '09
 ShengJyhWang

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