Think Bayes

The Monty Hall problem

To solve the Monty Hall problem, I’ll define a new class:

class Monty(Pmf):

    def __init__(self, hypos):
        Pmf.__init__(self)
        for hypo in hypos:
            self.Set(hypo, 1)
        self.Normalize()

So far Monty and Cookie are exactly the same. And the code that creates the Pmf is the same, too, except for the names of the hypotheses:

    hypos = 'ABC'
    pmf = Monty(hypos)

Calling Update is pretty much the same:

    data = 'B'
    pmf.Update(data)

And the implementation of Update is exactly the same:

    def Update(self, data):
        for hypo in self.Values():
            like = self.Likelihood(data, hypo)
            self.Mult(hypo, like)
        self.Normalize()

The only part that requires some work is Likelihood:

    def Likelihood(self, data, hypo):
        if hypo == data:
            return 0
        elif hypo == 'A':
            return 0.5
        else:
            return 1

Finally, printing the results is the same:

    for hypo, prob in pmf.Items():
        print hypo, prob

And the answer is

A 0.333333333333
B 0.0
C 0.666666666667

In this example, writing Likelihood is a little complicated, but the framework of the Bayesian update is simple. The code in this section is available from http://thinkbayes.com/monty.py. For more information see Working with the code.