Two Words, One Posterior

A Bernoulli naive Bayes spam filter has $p(\text{spam}) = 0.3$ and these fitted conditionals:

feature	$p(\cdot \mid \text{spam})$	$p(\cdot \mid \text{ham})$
contains "free"	0.6	0.1
contains "meeting"	0.1	0.5

An email contains both words. Using $p(y \mid x) \propto p(y) \prod_j p(x_j \mid y)$ , what is $p(\text{spam} \mid x)$ ?

Undefined without $p(x)$ , since the denominator needs the joint distribution of the two words, which the naive Bayes assumption refuses to model.

About 0.34: the class scores are $0.3(0.6)(0.1) = 0.018$ and $0.7(0.1)(0.5) = 0.035$ , and $0.018/(0.018 + 0.035) = 0.34$ .

About 0.40: average the two single-feature posteriors, 0.72 for "free" and 0.08 for "meeting", since the features are treated independently.

0.6, which is $p(\text{free} \mid \text{spam})$ : the model reads off the strongest piece of evidence.