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==Maximum Likelihood Estimation (ML)== | ==Maximum Likelihood Estimation (ML)== | ||
| + | <math>\hat a_{ML} = \text{max}_a ( f_{X}(x_i;a))</math> continuous | ||
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| + | <math>\hat a_{ML} = \text{max}_a ( Pr(x_i;a))</math> discrete | ||
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| + | If X is a binomial (n,p), where is X is number of heads n tosses, | ||
| + | Then, for any fixed k-value; | ||
| + | |||
| + | <math>\hat p_{ML}(k) = k/n</math> | ||
| + | |||
| + | If X is exponential then it's ML estimate is: | ||
| + | |||
| + | <math> \frac{1}{ \overline{X}} </math> | ||
==Maximum A-Posteriori Estimation (MAP)== | ==Maximum A-Posteriori Estimation (MAP)== | ||
Revision as of 04:06, 12 December 2008
Contents
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
If X is a binomial (n,p), where is X is number of heads n tosses,
Then, for any fixed k-value;
$ \hat p_{ML}(k) = k/n $
If X is exponential then it's ML estimate is:
$ \frac{1}{ \overline{X}} $
Maximum A-Posteriori Estimation (MAP)
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) $
Mean square error : $ 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]) $
Hypothesis Testing: ML Rule
Type I error
Type II error
Hypothesis Testing: MAP Rule
Overall P(err)
