2. Consider a Bayesian classification problem where we wish to determine if a contact on a social network is a friend, relative, or co-worker. We use four features - x₁ (MessageFrequency), x₂ (MainMessageType), x₃ (Proximity), x₄ Age. Each feature takes on one of up to five values, shown below:

MessageFrequency	Annually	Monthly	Daily
MainMessageType	StatusUpdate	DirectMessage
Proximity	SameCity	SameCountry	SameState
Age	Child	SeniorCitizen	MiddleAge	Teen

We classify each user as one of three classes: yⁱ=Friend, yⁱ=Relative, yⁱ=Co-worker. Based on a large training set, we wish to estimate all joint probability likelihoods, e.g.,

P(x₁=Monthly, x₂=StatusUpdate, x₃=SameBuilding, x₄=Teen | y=Co-Worker),

P(x₁=Daily, x₂=StatusUpdate, x₃=SameState, x₄=SeniorCitizen| y=Co-Worker).

as well as the class priors

a) Assuming the features are independent, how many total parameters need to be estimated, accounting for classifying friends, relatives, and co-workers?

b) Assuming the features are not independent, how many total parameters need to be estimated, accounting for classifying friends, relatives, and co-workers?

Now assume we replace the final discrete feature values with numbers:

Age

0(Child)

1(Teen)

2(MiddleAge)

3(SeniorCitizen)

We use a Gaussian likelihood for the probability of this feature for each class y: P(x_Age | y).

c) Assuming all features are independent, how many parameters need to be learned to compute the posterior probability P(y | x_age, x_proxim, x_msgType, x_msgFreq)