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Then <math>Z_n = \frac{\Sigma_{i=1}^n X_i - \Sigma_{i=1}^n \mu_i} {\sqrt{\Sigma_{i=1}^n \sigma^2}}</math> has <math>P(Z_n)\longrightarrow N(\mu ,\Sigma)</math> when <math>n \longrightarrow \infty</math> | Then <math>Z_n = \frac{\Sigma_{i=1}^n X_i - \Sigma_{i=1}^n \mu_i} {\sqrt{\Sigma_{i=1}^n \sigma^2}}</math> has <math>P(Z_n)\longrightarrow N(\mu ,\Sigma)</math> when <math>n \longrightarrow \infty</math> | ||
| + | |||
| + | ------------------------------------- | ||
| + | Error bounds for Bayes decision rule: | ||
| + | |||
| + | As we know Bayes decision rule guarantees the lowest average error rate; It Does not tell what the probability of error actually is. Therefore people try to find upper bounds for that: Chernoff and Bhattacharyya bounds | ||
| + | |||
| + | <math>P(error)=\int_{R^n} p(error,x)dx = \int_{R^n} p(error|x)p(x)dx</math> | ||
| + | |||
| + | in two class case we have: | ||
| + | |||
| + | <math> p(error|x) = min \{ p(\omega_1|x) , p(\omega_2|x) \} </math> | ||
| + | <math> \Rightarrow P(error)= \int_{R^n}min\{p(\omega_1|x),p(\omega_2|x)\} p(x)dx </math> | ||
| + | |||
| + | as we know from Lemma: <math>min \{a,b\}\leq a^\beta b^ {1-\beta}</math>, <math>\forall a,b \geq 0 , \forall \beta s.t 0 \leq\beta\leq 1 </math> | ||
Revision as of 12:33, 9 May 2010
Classic central limit Thm (Second Fundamental probabilistic):
"The distribution of the average of a large number of samples from a distribution tends to be normal"
let X1,X2,...,Xn be n independent and identically distributed variables (i.i.d) with finite mean $ \mu $ and finite variance $ \sigma^2>0 $.Then as n increases the distribution of $ \Sigma_{i=1}^n \frac{X_i} {n} $ approaches $ N(\mu,\frac {\sigma^2}{n}) $.
More precisely the random variable $ Z_n = \frac{\Sigma_{i=1}^n X_i - n \mu}{\sigma \sqrt{n}} $ has $ P(Z_n)\longrightarrow N(0,1) $ when $ n \longrightarrow \infty $
More generalization of central limit Thm.
let X1,X2,...,Xn be n independent variables
Xi has mean $ \mu_i $ & finite variance $ \sigma^2 > 0 $ ,i=1,2,...,n
Then $ Z_n = \frac{\Sigma_{i=1}^n X_i - \Sigma_{i=1}^n \mu_i} {\sqrt{\Sigma_{i=1}^n \sigma^2}} $ has $ P(Z_n)\longrightarrow N(\mu ,\Sigma) $ when $ n \longrightarrow \infty $
Error bounds for Bayes decision rule:
As we know Bayes decision rule guarantees the lowest average error rate; It Does not tell what the probability of error actually is. Therefore people try to find upper bounds for that: Chernoff and Bhattacharyya bounds
$ P(error)=\int_{R^n} p(error,x)dx = \int_{R^n} p(error|x)p(x)dx $
in two class case we have:
$ p(error|x) = min \{ p(\omega_1|x) , p(\omega_2|x) \} $ $ \Rightarrow P(error)= \int_{R^n}min\{p(\omega_1|x),p(\omega_2|x)\} p(x)dx $
as we know from Lemma: $ min \{a,b\}\leq a^\beta b^ {1-\beta} $, $ \forall a,b \geq 0 , \forall \beta s.t 0 \leq\beta\leq 1 $
