Exam correction

Intelligent Systems 2016-1

After solving all the questions in the exam save your notebook with the name username.ipynb and submit it to: https://www.dropbox.com/request/GM4PK3ibNSH46IejnzLP

1.

Consider the following Bayes net:

where $A$, $B$ and $x_i$ are all binary variables.

Write a function that takes as input a set of samples and calculates the parameter of the Bayes net model.



In [20]:

    
def bn_model(data, k):
    '''
    data: training data as a list of lists
    [[x_1, x_2, ..., X_n, A, B]
     [x_1, x_2, ..., X_n, A, B]
     :
     [x_1, x_2, ..., X_n, A, B]
    ]
    
    k: Laplace's smoothing parameter
    
    returns:
    It must return the model as a dictionary with the following form:
    For example:
    { 'A' : 0.1,
      ('B', 'A', 0) : 0.4
      ('B', 'A', 1) : 0.7
      ('x1', 'B', 0) : 0.3,
      ('x1', 'B', 1) : 0.1,
      ('x2', 'B', 0) : 0.2,
      ('x2', 'B', 1) : 0.6,
    }
    In this case the entry for 'A' means that P(A = 1) = 0.4 and therefore P(A = 0) = 0.6.
    In the same way, the entry for ('x1', 'B', 1) indicates that P(X_1 = 1 | B = 1) = 0.1 and
    therefore P(X_1 = 0 | B = 1) = 0.9.
    '''
    model = {}
    return model

data = [[0, 0, 1, 1], 
        [0, 1, 0, 0], 
        [0, 0, 0, 0], 
        [1, 1, 0, 1], 
        [0, 1, 1, 1], 
        [0, 1, 0, 1]]


bn_model(data,3.5)









    Out[20]:





{}

2.

Write a function that given a model calculates $P(B|x_1,\dots,x_n)$



In [12]:

    
def p1(model, X):
    '''
    model: a dictionary with the model probabilities.
    X: a list with x_i values [x_1, x_2, ... , x_n]
    Returns: the probability P(B = 1 | x_1, x_2, ... , x_n)
    '''
    
    return 0.0

3.

Write a function that given a model calculates $P(A|x_1,\dots,x_n)$



In [13]:

    
def p2(model, X):
    '''
    model: a dictionary with the model probabilities.
    X: a list with x_i values [x_1, x_2, ... , x_n]
    Returns: the probability P(A = 1 | x_1, x_2, ... , x_n)
    '''
    return 0.0

4.

Write a function that given a model calculates $P(A|x_1, x_n)$



In [14]:

    
def p3(model, x_1, x_n):
    '''
    model: a dictionary with the model probabilities.
    x_1, x_n: x values
    Returns: the probability P(A = 1 | x_1, x_n)
    '''
    return 0.0



In [21]:

    
""" Testing """

def approx_equal(val1, val2):
    return abs(val1-val2) <= 0.00001

def test_dict(test,answer):
    if sorted(test.keys()) != sorted(answer.keys()): return False

    for k,v in test.items():
        if not approx_equal(v,answer[k]):
            return False
        
    return True


def test_P1():
    data = [[0, 0, 1, 0], 
            [0, 0, 1, 0], 
            [0, 0, 0, 0], 
            [1, 0, 0, 1], 
            [0, 0, 1, 1], 
            [0, 0, 0, 1]]

    student = bn_model(data,0.0)
    print student
    answer = {'A': 0.5, ('x1', 'B', 0): 0.0, ('x2', 'B', 0): 0.0, ('B', 'A', 0): 0.6666666666666666, ('x2', 'B', 1): 0.0, ('B', 'A', 1): 0.3333333333333333, ('x1', 'B', 1): 0.3333333333333333}    
    assert(test_dict(student,answer))
    print "Test 1"
    
    
    data = [[0, 0, 1, 1], 
            [0, 1, 0, 0], 
            [0, 0, 0, 0], 
            [1, 1, 0, 1], 
            [0, 1, 1, 1], 
            [0, 1, 0, 1]]


    student = bn_model(data,0.0)
    print student
    answer = {'A': 0.3333333333333333, ('x1', 'B', 0): 0.0, ('x2', 'B', 0): 0.5, ('B', 'A', 0): 0.5, ('x2', 'B', 1): 0.75, ('B', 'A', 1): 1.0, ('x1', 'B', 1): 0.25}
    assert(test_dict(student,answer))
    print "Test 2"
    
    data = [[0, 0, 1, 0], 
            [0, 0, 1, 0], 
            [0, 0, 0, 0], 
            [1, 0, 0, 1], 
            [0, 0, 1, 1], 
            [0, 0, 0, 1]]
    
    student = bn_model(data,1.0)
    print student
    answer = {'A': 0.5,
             ('B', 'A', 0): 0.6,
             ('B', 'A', 1): 0.4,
             ('x1', 'B', 0): 0.2,
             ('x1', 'B', 1): 0.4,
             ('x2', 'B', 0): 0.2,
             ('x2', 'B', 1): 0.2}
    assert(test_dict(student,answer))
    print "Test 3"
    
    
    data = [[0, 0, 1, 1], 
            [0, 1, 0, 0], 
            [0, 0, 0, 0], 
            [1, 1, 0, 1], 
            [0, 1, 1, 1], 
            [0, 1, 0, 1]]


    student = bn_model(data,3.5)
    print student
    answer = {'A': 0.4230769230769231,
             ('B', 'A', 0): 0.5,
             ('B', 'A', 1): 0.6111111111111112,
             ('x1', 'B', 0): 0.3888888888888889,
             ('x1', 'B', 1): 0.4090909090909091,
             ('x2', 'B', 0): 0.5,
             ('x2', 'B', 1): 0.5909090909090909}
    assert(test_dict(student,answer))
    print "Test 4"

        
def test_P2():

    model = {  'A' : 0.35,
              ('B', 'A', 0) : 0.2,
              ('B', 'A', 1) : 0.7,
              ('x1', 'B', 1) : 0.44,
              ('x1', 'B', 0) : 0.74,
        }
    
    #print p1(model,[1])
    assert(approx_equal(0.262948207171,p1(model,[1])))

    model = {  'A' : 0.35,
                  ('B', 'A', 0) : 0.2,
                  ('B', 'A', 1) : 0.7,
                  ('x1', 'B', 1) : 0.44,
                  ('x1', 'B', 0) : 0.74,
                  ('x2', 'B', 0) : 0.84,
                  ('x2', 'B', 1) : 0.5,
            }

    #print p1(model,[1,0])
    assert(approx_equal(0.527156549521,p1(model,[1,0])))


    model = {  'A' : 0.35,
                  ('B', 'A', 0) : 0.2,
                  ('B', 'A', 1) : 0.7,
                  ('x1', 'B', 1) : 0.44,
                  ('x1', 'B', 0) : 0.74,
                  ('x2', 'B', 0) : 0.84,
                  ('x2', 'B', 1) : 0.5,
                  ('x3', 'B', 1) : 0.1,
                  ('x3', 'B', 0) : 0.2,
            }
    
    #print p1(model,[1,0,1])
    assert(approx_equal(0.357917570499,p1(model,[1,0,1])))

    model = {      'A' : 0.35,
                  ('B', 'A', 0) : 0.2,
                  ('B', 'A', 1) : 0.7,
                  ('x1', 'B', 1) : 0.44,
                  ('x1', 'B', 0) : 0.74,
                  ('x2', 'B', 0) : 0.84,
                  ('x2', 'B', 1) : 0.5,
                  ('x3', 'B', 1) : 0.1,
                  ('x3', 'B', 0) : 0.2,
                  ('x4', 'B', 1) : 0.1,
                  ('x4', 'B', 0) : 0.9
            }

    #print p1(model,[1,0,1,1])
    assert(approx_equal(0.0583244962884,p1(model,[1,0,1,1])))



def test_P3():
    
    model = {  'A' : 0.35,
              ('B', 'A', 0) : 0.2,
              ('B', 'A', 1) : 0.7,
              ('x1', 'B', 1) : 0.44,
              ('x1', 'B', 0) : 0.74,
        }
    
    print p2(model,[1])
    assert(approx_equal(0.29561752988,p2(model,[1])))
    
    model = {  'A' : 0.35,
                  ('B', 'A', 0) : 0.2,
                  ('B', 'A', 1) : 0.7,
                  ('x1', 'B', 1) : 0.44,
                  ('x1', 'B', 0) : 0.74,
                  ('x2', 'B', 0) : 0.84,
                  ('x2', 'B', 1) : 0.5,
            }
    
    #print p2(model,[1,0])
    assert(approx_equal(0.423846645367,p2(model,[1,0])))
    
    
    model = {  'A' : 0.35,
                  ('B', 'A', 0) : 0.2,
                  ('B', 'A', 1) : 0.7,
                  ('x1', 'B', 1) : 0.44,
                  ('x1', 'B', 0) : 0.74,
                  ('x2', 'B', 0) : 0.84,
                  ('x2', 'B', 1) : 0.5,
                  ('x3', 'B', 1) : 0.1,
                  ('x3', 'B', 0) : 0.2,
            }
    
    #print p2(model,[1,0,1])
    assert(approx_equal(0.341709327549,p2(model,[1,0,1])))

    model = {      'A' : 0.35,
                  ('B', 'A', 0) : 0.2,
                  ('B', 'A', 1) : 0.7,
                  ('x1', 'B', 1) : 0.44,
                  ('x1', 'B', 0) : 0.74,
                  ('x2', 'B', 0) : 0.84,
                  ('x2', 'B', 1) : 0.5,
                  ('x3', 'B', 1) : 0.1,
                  ('x3', 'B', 0) : 0.2,
                  ('x4', 'B', 1) : 0.1,
                  ('x4', 'B', 0) : 0.9
            }

    #print p2(model,[1,0,1,1])
    assert(approx_equal(0.196306822199,p2(model,[1,0,1,1])))

    
def test_P4():
    
    model = {  'A' : 0.35,
                  ('B', 'A', 0) : 0.2,
                  ('B', 'A', 1) : 0.7,
                  ('x1', 'B', 1) : 0.44,
                  ('x1', 'B', 0) : 0.74,
                  ('x2', 'B', 0) : 0.84,
                  ('x2', 'B', 1) : 0.5,
                  ('x3', 'B', 1) : 0.1,
                  ('x3', 'B', 0) : 0.2,
            }
    
    assert(approx_equal(0.358504672897 ,p3(model,0,1)))

    model = {  'A' : 0.35,
                  ('B', 'A', 0) : 0.2,
                  ('B', 'A', 1) : 0.7,
                  ('x1', 'B', 1) : 0.44,
                  ('x1', 'B', 0) : 0.74,
                  ('x2', 'B', 0) : 0.84,
                  ('x2', 'B', 1) : 0.5,
                  ('x3', 'B', 1) : 0.1,
                  ('x3', 'B', 0) : 0.2,
                  ('x4', 'B', 1) : 0.1,
                  ('x4', 'B', 0) : 0.9
            }

    
    
    assert(approx_equal(0.186504910456,p3(model,1,1)))


model = {  'A' : 0.35,
                  ('B', 'A', 0) : 0.2,
                  ('B', 'A', 1) : 0.7,
                  ('x1', 'B', 1) : 0.44,
                  ('x1', 'B', 0) : 0.74,
                  ('x2', 'B', 0) : 0.84,
                  ('x2', 'B', 1) : 0.5,
                  ('x3', 'B', 1) : 0.1,
                  ('x3', 'B', 0) : 0.2,
                  ('x4', 'B', 1) : 0.1,
                  ('x4', 'B', 0) : 0.9
            }


test_P1()









    



{}






    



---------------------------------------------------------------------------
AssertionError                            Traceback (most recent call last)
<ipython-input-21-9be4624eec68> in <module>()
    244 
    245 
--> 246 test_P1()

<ipython-input-21-9be4624eec68> in test_P1()
     25     print student
     26     answer = {'A': 0.5, ('x1', 'B', 0): 0.0, ('x2', 'B', 0): 0.0, ('B', 'A', 0): 0.6666666666666666, ('x2', 'B', 1): 0.0, ('B', 'A', 1): 0.3333333333333333, ('x1', 'B', 1): 0.3333333333333333}
---> 27     assert(test_dict(student,answer))
     28     print "Test 1"
     29 

AssertionError:



In [ ]: