You are given the following incomplete function. The goal is to complete it, so that the conditional probabilities P([x]_a|y) in naivebayes is estimated. Our features are binary categorical. Use a categorical distribution and return the prob vectors for each feature being 1 given a class label. Note that the result will be two vectors of length d (the number of features), where the values represent the probability that feature i is equal to 1.

def naivebayesPXY(X,Y):

"""

naivebayesPXY(X, Y) returns [posprob,negprob]

Input:

X : n input vectors of d dimensions (nxd)

Y : n labels (-1 or +1) (n)

Output:

posprob: probability vector of p(x_alpha = 1|y=1)(d)

negprob: probability vector of p(x_alpha = 1|y=-1) (d)

"""

# add one positive and negative example to avoid division by zero ("plus-one smoothing")

n, d = X.shape

X = np.concatenate([X, np.ones((2,d)), np.zeros((2,d))])

Y = np.concatenate([Y, [-1,1,-1,1]])

n, d = X.shape

# YOUR CODE HERE

raise NotImplementedError()

posprob, negprob = naivebayesPXY(X,Y)

To test that you have completed this correctly, the following should all run.

# The following tests check that your implementation of naivebayesPXY returns the same posterior probabilities as the correct implementation, in the correct dimensions

# test a simple toy example with two points (one positive, one negative)

def naivebayesPXY_test1():

x = np.array([[0,1],[1,0]])

y = np.array([-1,1])

pos, neg = naivebayesPXY(x,y)

pos0, neg0 = naivebayesPXY_grader(x,y)

test = np.linalg.norm(pos - pos0) + np.linalg.norm(neg - neg0)

return test < 1e-5

# test the probabilities P(X|Y=+1)

def naivebayesPXY_test2():

pos, neg = naivebayesPXY(X,Y)

posprobXY, negprobXY = naivebayesPXY_grader(X, Y)

test = np.linalg.norm(pos - posprobXY)

return test < 1e-5

# test the probabilities P(X|Y=-1)

def naivebayesPXY_test3():

pos, neg = naivebayesPXY(X,Y)

posprobXY, negprobXY = naivebayesPXY_grader(X, Y)

test = np.linalg.norm(neg - negprobXY)

return test < 1e-5

# Check that the dimensions of the posterior probabilities are correct

def naivebayesPXY_test4():

pos, neg = naivebayesPXY(X,Y)

posprobXY, negprobXY = naivebayesPXY_grader(X, Y)

return pos.shape == posprobXY.shape and neg.shape == negprobXY.shape

runtest(naivebayesPXY_test1,'naivebayesPXY_test1')

runtest(naivebayesPXY_test2,'naivebayesPXY_test2')

runtest(naivebayesPXY_test3,'naivebayesPXY_test3')

runtest(naivebayesPXY_test4,'naivebayesPXY_test4')