The Word That Cancels the Case

A Naive Bayes spam filter is trained on a corpus where $P(\text{Spam}) = 0.25$ and $P(\text{Ham}) = 0.75$ .

An incoming email contains exactly three indicator words: invoice, urgent, and kitten. The word kitten never appeared in a single spam email during training, so the unsmoothed estimate would have been $P(\text{kitten} \mid \text{Spam}) = 0$ . After Laplace smoothing the filter's estimates are:

word	$P(w \mid \text{Spam})$	$P(w \mid \text{Ham})$
invoice	$0.40$	$0.20$
urgent	$0.50$	$0.10$
kitten	$0.02$	$0.20$

The classifier assumes the three words are conditionally independent given the class.

What is $P(\text{Spam} \mid \textit{invoice}, \textit{urgent}, \textit{kitten})$ ?

The Word That Cancels the Case

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