The table shows the examples of SPAM and those of HAM messages which are consisted of some words whose dictionary size is twelve. Suppose that youve received SPAM messages for the 1^st 3 days, then HAM messages for the next 5 days, i.e. a data as a sequence of message is .

SPAM	HAM
offer is secret	play golf tomorrow
click secret link	went play golf
secret golf link	secret golf event
	golf is tomorrow
	golf costs money

(a) Compute the maximum likelihood of SPAM, i.e. P(SPAM)=q, using a log-likelihood.

(b) In the Bayesian network of this ML parameter learning,

(i) Draw the BN with the CPT of the required parameters (e.g. q₁, q₂, ..). -- You dont have to compute the exact values of parameters yet.

(ii) How many parameters are required?

(c) By ML-learning, compute a parameter value, P(secret | SPAM) and P(secret | HAM), respectively.

(d) Now, the new message golf is received. What is the probability that this message is SPAM?

(e) The new message secret is secret is received. What is the probability that this is SPAM?

(f) For a new message, tomorrow is secret, what is the probability that the message is SPAM and HAM, respectively?