Bayesian Filters: An unwanted and unsolicited email is commonly referred to as spam. By contrast, an email that's not spam is sometimes referred to as ham. Email providers try their best to identify and filter out spam emails. One way of doing this is to use a Bayesian filter. By collecting vast amounts of emails identified as either ham or spam, an email provider can create a list of words that commonly show up only in spam. A Bavesian filter is called unbiased if it assumes that any incoming email is equally likely to be ham or spam.

$50-50$

Suppose a particular provider puts together a list of words for which their collected data says the following:

$90\%$ of spam emails contain at least $2$ words from the list, and

ham

filter

$80\%$ of ham emails do not contain at least $2$ words from the list. unbiased

The provider intends to use this spam$-$likely word list as the basis for an unbiased Bayesian filter. If an incoming email does contain at least $2$ words from the list, what is the probability that the email is spam?

spam $||$ at least $2$ words$)$:

Exploration: Suppose the email provider uses Bayes' Theorem as you just did above and they decide the resulting spam $||$ at least $2$ words$)$ is not a high enough probability. They're worried about their Bayesian filter accidentally putting important ham emails into the spam folder. So they design a second Bayesian filter. This time they notice that, in their collected data:

$60\%$ of spam emails contain a typo in a name, and

$70\%$ of ham emails do not contain a typo in a name.

The provider intends to use this information to create a second Bayesian filter, but this one is not unbiased. This filter will only be used on an email that has gone through the previous filter and been found to contain at least $2$ words from the spam$-$likely word list. If an email gets caught in both filters $($i$.$e$.,$ it has at least $2$ spam$-$likely words and it has a typo in a name$),$ then what is the probability that the email is spam?

Does it matter which filter is applied first?

spam $||$ at least $2$ words and a typo in a name$)$: