(Likelihood Ratio TEST)
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Choose threshold  (T),
 
Choose threshold  (T),
  
say H_1 if L(x) > T
+
say <math>\ H_{\rm 1}  ;if L(x) > T</math>
  
say H_0 if L(x) < T
+
say <math>\ H_{\rm 0}  ;if L(x) < T</math>
  
 
so ML Rule is an LRT with T = 1
 
so ML Rule is an LRT with T = 1
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so ML Rule is an LRT with T =1
 
so ML Rule is an LRT with T =1
 +
 
== '''Law Of Iterated Expectation''' ==
 
== '''Law Of Iterated Expectation''' ==
  

Revision as of 12:47, 13 December 2008

Maximum Likelihood Estimation (ML)

$ \hat a_{ML} = \text{max}_a ( f_{X}(x_i;a)) $ continuous

$ \hat a_{ML} = \text{max}_a ( Pr(x_i;a)) $ discrete

Maximum A-Posteriori Estimation (MAP)

$ \hat \theta_{MAP}(x) = \text{arg max}_\theta P_{X|\theta}(x|\theta)P_ {\theta}(\theta) $

$ \hat \theta_{MAP}(x) = \text{arg max}_\theta f_{X|\theta}(x|\theta)P_ {\theta}(\theta) $

Minimum Mean-Square Estimation (MMSE)

$ \hat{y}_{\rm MMSE}(x) = \int\limits_{-\infty}^{\infty}\ {y}{f}_{\rm Y|X}(y|x)\, dy={E}(Y|X=x) $

Likelihood Ratio TEST

How to find a good rule? --Khosla 16:44, 13 December 2008 (UTC)

$ \ L(x) = P_{\rm X|\theta} (x|\theta1) / P_{\rm X|\theta} (x|\theta1) $

Choose threshold (T),

say $ \ H_{\rm 1} ;if L(x) > T $

say $ \ H_{\rm 0} ;if L(x) < T $

so ML Rule is an LRT with T = 1

as T increases Type I Error Increases

as T increases Type II Error Decreases

& Vice Versa

so ML Rule is an LRT with T =1

Law Of Iterated Expectation

Unconditional Expectaion--E[X] = E{E[x|theta]}--Umang 16:10, 13 December 2008 (UTC)umang


Mean square error :

Headline text

$ MSE = E[(\theta - \hat \theta(x))^2] $

Linear Minimum Mean-Square Estimation (LMMSE)

$ \hat{y}_{\rm LMMSE}(x) = E[\theta]+\frac{COV(x,\theta)}{Var(x)}*(x-E[x]) $

Law of Iterated Expectation: E[E[X|Y]]=E[X]

Hypothesis Testing: ML Rule

Given a value of X, we will say H1 is true if X is in region R, else will will say H0 is true.

Type I error

Say H1 when truth is H0. Probability of this is: Pr(Say H1|H0) = Pr(X is in R|theta0)

Type II error

Say H0 when truth is H1. Probability of this is: Pr(Say H0|H1) = Pr(X is NOT in R|theta1)


Hypothesis Testing: MAP Rule

Overall P(err) = $ P_{\theta}(\theta_{0})Pr[Say H_{1}|H_{0}]+P_{\theta}(\theta_{1})Pr[Say H_{0}|H_{1}] $

Alumni Liaison

has a message for current ECE438 students.

Sean Hu, ECE PhD 2009