User:Herdrick

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[edit] Naive Bayes

We want to decide how like it is that a message is spam given that it contains the words "free" and "viagra".

$\Pr(\mathrm{spam}|\mathrm{free,viagra}) = \frac{\Pr(\mathrm{free,viagra}|\mathrm{spam})\Pr(\mathrm{spam})}{\Pr(\mathrm{free,viagra})}$

Since we assume independence of the variables, we can rewrite it like so:

$\Pr(\mathrm{spam}|\mathrm{free,viagra}) = \frac{\Pr(\mathrm{free}|\mathrm{spam})\Pr(\mathrm{viagra}|\mathrm{spam})\Pr(\mathrm{spam})}{\Pr(\mathrm{free})\Pr(\mathrm{viagra})}$

Which is identical to this:

$\Pr(\mathrm{spam}|\mathrm{free,viagra}) = \Pr(\mathrm{spam})\frac{\Pr(\mathrm{free}|\mathrm{spam})\Pr(\mathrm{viagra}|\mathrm{spam})}{\Pr(\mathrm{free})\Pr(\mathrm{viagra})}$