Contents
Maximum Likelihood Estimation (ML)
- $ \hat a_{ML} = \overset{max}{a} f_{X}(x_i;a) $ continuous
- $ \hat a_{ML} = \overset{max}{a} Pr(x_i;a) $ discrete
Chebyshev Inequality
"Any RV is likely to be close to its mean"
- $ \Pr(\left|X-E[X]\right|\geq C)\leq\frac{var(X)}{C^2}. $
Maximum A-Posteriori Estimation (MAP)
- $ \hat \theta_{MAP}(x) = \text{arg }\overset{max}{\theta} P_{X|\theta}(x|\theta)P_ {\theta}(\theta) $
- $ \hat \theta_{MAP}(x) = \text{arg }\overset{max}{\theta} f_{X|\theta}(x|\theta)P_ {\theta}(\theta) $
Minimum Mean-Square Estimation (MMSE)
- $ \hat{y}_{\rm MMSE}(x) = \int_{-\infty}^{\infty} {y}{f}_{\rm Y|X}(y|x)\, dy={E}[Y|X=x] $
Law Of Iterated Expectation
- $ E[E[X|Y]] = \begin{cases} \sum_{y} E[X|Y = y]p_Y(y),\,\,\,\,\,\,\,\,\,\,\mbox{ Y discrete,}\\ \int_{-\infty}^{+\infty} E[X|Y = y]f_Y(y)\,dy,\mbox{ Y continuous.} \end{cases} $
Using the total expectation theorem:
- $ E\Big[ E[X|Y]] = E[X] $
Mean Square Error
- $ MSE = E[(\Theta - \hat \theta(x))^2] $
- $ MSE(E(\Theta)) = var(\Theta) \, $
Linear Minimum Mean-Square Estimation (LMMSE)
The LMMS estimator $ \hat{Y} $ of Y based on the variable X is
- $ \hat{Y}_{LMMSE}(x) = E[Y]+\frac{COV(Y,X)}{Var(X)}(X-E[X]) = E[Y] + \rho \frac{\sigma_{Y}}{\sigma_{X}}(X-E[X]) $
where
- $ \rho = \frac{COV(Y,X)}{\sigma_{Y}\sigma_{X}} $
Law of Iterated Expectation: E[E[X|Y]]=E[X]
COV(X,Y)=E[XY] - E[X]E[Y]
Hypothesis Testing
In hypothesis testing $ \Theta $ takes on one of m values, $ \theta_1,...,\theta_m $ where m is usually small; often m = 2, in which case it is a binary hypthothesis testing problem.
The event $ \Theta = \theta_i $ is the $ i^{th} $ hypothesis denoted by $ H_i $
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: False Rejection
Say $ H_1 $ when truth is $ H_0 $. Probability of this is:
- $ Pr(\mbox{Say } H_1|H_0) = Pr(x \in R|\theta_0) $
Type II Error: False Acceptance
Say $ H_0 $ when truth is $ H_1 $. Probability of this is:
- $ Pr(\mbox{Say }H_0|H_1) = Pr(x \in R^C|\theta_1) $
Say H1 if;
- $ \{f_{X|\theta}(x|\theta1) $ > $ \{f_{X|\theta}(x|\theta0) $
Else H0
Say H0 if;
- $ \{f_{X|\theta}(x|\theta1) $ <= $ \{f_{X|\theta}(x|\theta0) $
Else H1
MAP Rule
- $ \mbox{Overall P(err)} = P_{\theta}(\theta_{0})Pr\Big[\mbox{Say }H_{1}|H_{0}\Big] +P_{\theta}(\theta_{1})Pr\Big[\mbox{Say }H_{0}|H_{1}\Big] $
Note that for Overall P(error), cannot use values from ML estimate.
Likelihood Ratio Test
How to find a good rule? --Khosla 16:44, 13 December 2008 (UTC)
For X is discrete
- $ \ L(x) = \frac{p_{X|\theta} (x|\theta_1)}{p_{X|\theta} (x|\theta_0)} $
Choose threshold (T),
- $ \mbox{Say } \begin{cases} H_{1}; \mbox{ if } L(x) > T\\ H_{0}; \mbox{ if } L(x) < T \end{cases} $
The Maximum Likelihood rule is a Likelihood Ratio Test with T = 1 The MAP rule is a Likelihood Ratio Test with $ T=\frac{P_\theta(\theta_0)}{P_\theta(\theta_1)} $
Observations:
- as T decreases Type I Error Increases
- as T decreases Type II Error Decreases
- as T increases Type I Error Decreases
- as T increases Type II Error Increases
($ T = 0 \Rightarrow R = \{x|P_{X|\theta}(x|\theta_1) > 0\} $. So, Type I error ($ Pr(x\in R | H_0) $) is maximized as T is minimized.)
The threshold value T=1, corresponds to the ML rule.
