Question

Question: Suppose that a Bayesian spam filter is trained on a set of 1000 spam messages and 400 messages that are not spam. The word “opportunity” appears in 175 spam messages and 2 messages that are not spam. Would an incoming message be rejected as spam if it contains the word “opportunity” and the threshold for rejecting a message is 0.9?

Short answer

Answer:

Incoming message would not be rejected as spam.

Step-by-step solution

Given

A Bayesian spam filter is trained on a set of 1000 spam messages and 400 messages that are not spam. The word “opportunity” appears in 175 spam messages and 2 messages that are not spam.

To find: If the incoming message be rejected as spam or not.

Formula

Bayes’ Formula:

\(P(F/E) = \frac{{P(E/F)P(F)}}{{P(E/F)P(F) + P(E/\bar F)P(\bar F)}}\)

Calculation

Consider the following events:

E = A received message contains the word ‘opportunity’.

\({{\rm{F}}_1}\)= an incoming message is spam

and

\({{\rm{F}}_2}\)= an incoming message is not spam

The probability that a message should be rejected as spam given it contains the word ‘opportunity’.

\(P\left( {{F_1}/E} \right) = \frac{{P\left( {E/{F_1}} \right)P\left( {{F_1}} \right)}}{{P\left( {E/{F_1}} \right)P\left( {{F_1}} \right) + P\left( {E/{F_2}} \right)P\left( {{F_2}} \right)}}\)

\(\begin{aligned}{l} &= \frac{{\left( {\frac{{175}}{{1000}}} \right)\left( {\frac{{1000}}{{1400}}} \right)}}{{\left( {\frac{{175}}{{1000}}} \right)\left( {\frac{{1000}}{{1400}}} \right) + \left( {\frac{{20}}{{400}}} \right)\left( {\frac{{400}}{{1400}}} \right)}}\\ &= \frac{{175}}{{195}}\\ &= 0.897 < 0.9\end{aligned}\)

So, this message would not be rejected as spam.

Final Answer

Thus, incoming message would not be rejected as spam.