Artificial Intelligence Question 1: Consider a Naive Bayes classifier for spam filtering

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Artificial Intelligence Question 1:

Consider a Naive Bayes classifier for spam filtering. We are given a training set of 500 randomly chosen emails. We examine them and label 200 of them as spam emails and 300 as non-spam emails. There are 2000 (words) in the 200 spam emails. 200 spam emails contain the word "a"; 60 contain the word "good"; and 50 contain the word "job". In the 300 nonspam emails, there are in total 1000 (words). 150 non-spam emails contain the word "a''; 30 non-spam emails contain the word "good'', and 10 non-spam emails contain the word "job''. We use S to denote a random event that one email is found to be spam, and use NS to denote non-spam. We use P(word|S) to denote the conditional probability that one word word appears in a spam email (P(word|NS) is defined similarly). P(word1, word2|S) is the probability that both word1 and word2 appear in a spam email.

i. What is the best approximation to P( "a"|S) and P( "good"|S) and P("job"|S) given the training set?

ii. What is the best approximation to P( "a", "good", "job"|S) and P( "a", "good", "job"|NS) given the training set (Hint: using the structure of a Naive Bayes network to answer this question)?

iii. Given a testing email "Well done! You did a good job in CS471!". Will a Naive Bayes classifier trained on the training set above classify it as a spam? Why or why not? (Hint: you should make the decision based on P(S| "a", "good", "job") and P(NS| "a", "good", "job").)