Exam

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/ppnwCHBo3G2eVo1tJPdY

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 [11]:

    
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.
    '''
    return {}

2.

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



In [ ]:

    
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

3.

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



In [12]:

    
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

4.

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



In [13]:

    
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