Exercise 3.22

Consider a Naive Bayes model (multivariate Bernoulli version) for spam
classification with the vocabulary V="secret", "offer", "low", "price",
"valued", "customer", "today", "dollar", "million", "sports", "is", "for",
"play", "healthy", "pizza".  We have the example spam messages "million dollar
offer", "secret offer today", "secret is secret" and normal messages, "low
price for valued customer", "play secret sports today", "sports is healthy",
"low price pizza".  Give the MLEs for the following parameters:
theta_spam, theta_secret|spam, theta_secret|non-spam, theta_sports|non-spam,
theta_dollar|spam.

The multivariate Bernoulli version uses indicators: the term is present or not present.

                                             customer    dollar    for    health    is    low    million    offer    pizza    play    price    secret    sports    today    valued
spam        million dollar offer                              1                                        1        1
            secret is secret                                                         1                                                              1
            secret offer today                                                                                  1                                   1                  1

                                             customer    dollar    for    health    is    low    million    offer    pizza    play    price    secret    sports    today    valued
non-spam    low price for valued customer           1                1                      1                                             1                                      1
            low price pizza                                                                 1                            1                1
            play secret sports today                                                                                             1                  1         1        1
            sports is healthy                                                  1     1                                                                        1

There are 3 spam messages and 4 non-spam messages.

theta_spam = 3 / (3 + 4) = 3 / 7
theta_secret|spam = 2 / (2 + 1) = 2 / 3
theta_secret|non-spam = 1 / (1 + 3) = 1 / 4
theta_sports|non-spam = 2 / (2 + 2) = 2 / 4 = 1 / 2
theta_dollar|spam = 1 / (1 + 2) = 1 / 3