Question

(a) You wish to build a na¨ıve Bayes classifier regressing Booleans A, B and C onto the Boolean X. Someone has already counted the data for you to create frequency tables below:

____A=0 A=1 B=0 B=1 C=0 C=1

X=0 10 40 30 20 15 35

X=1 30 20 5 45 40 10

a) Construct probability tables as needed to specify the estimated na¨ıve Bayes classifier for the task. Then give the formula for the classifier and describe how it would be used.

(b) Consider the probabilities p(A=0|X =0) and p(B=0|X =1). Compute their standard errors, making any assumptions as needed? What can you say about the resulting estimates?

Answer 1

a) The Naive Bayes classifier formula is given by:

P(A|B) = P(A) * P(B|A)/P(B)

b) P(A=0|X=0) = P(B=0|X=1)

P(A=0|X=0) = P(A=0) * P(X=0|A)/P(X=0)

P(B=0|X=1) = P(B=0) * P(X=0|B)/P(X=1)

Standard deviation for the sample, sigma = 14.35

Standard error = sigma/sqrt(n) = 14.35/sqrt(7) = 14.35/2.64 = 5.4360

The resulting estimates are good.