Question

Suppose that 8% of emails is spam and 92% (prior probabilities) are normal. The probabilities (likelihood of evidence) of occurrence of various worlds in normal and spam emails are given in the following table: word P(word|spam) P(word|normal) abandoned fund 0.5 0.01 deceased customer 0.6 0.05 Bank account 0.2 0.1

Consider the following email message “I am Mrs Sarah Boardman. I have decided to seek a confidential co-operation with you, During the course of our bank year auditing, I discovered an abandoned fund, sum total of US $3.5 Million in the bank account that belongs to a deceased customer who unfortunately lost his life and entire family in fatal gassy car accident. Reply me for more clarification if you are interested”.

Calculate the posterior probability that this message is spam.

Answer 1

The prior probability of a mail being spam is P(spam) = 0.08 and a mail being normal P(Normal) = 0.92

Let

be the event that the word "abandoned fund" occurs in a mail.

be the event that the word "deceased customer" occurs in a mail.

be the event that the word "Bank account" occurs in a mail.

We know the following conditional probabilities

The message contains all the 3 words

“I am Mrs Sarah Boardman. I have decided to seek a confidential co-operation with you, During the course of our bank year auditing, I discovered an abandoned fund, sum total of US $3.5 Million in the bank account that belongs to a deceased customer who unfortunately lost his life and entire family in fatal gassy car accident. Reply me for more clarification if you are interested”.

The joint probability of finding the 3 words given that the mail is spam is

The joint probability of finding the 3 words given that the mail is normal is

The unconditional joint probability is

Using the Bayes rule the posterior probability that the message is spam is

ans: the posterior probability that this message is spam is 0.9905