Think Bayes

Chapter 3. Estimation

The dice problem

Suppose I have a box of dice that contains a 4-sided die, a 6-sided die, an 8-sided die, a 12-sided die, and a 20-sided die. If you have ever played Dungeons & Dragons, you know what I am talking about.

Suppose I select a die from the box at random, roll it, and get a 6. What is the probability that I rolled each die?

Let me suggest a three-step strategy for approaching a problem like this.

1. Choose a representation for the hypotheses.

2. Choose a representation for the data.

3. Write the likelihood function.

In previous examples I used strings to represent hypotheses and data, but for the die problem I’ll use numbers. Specifically, I’ll use the integers 4, 6, 8, 12, and 20 to represent hypotheses:

    suite = Dice([4, 6, 8, 12, 20])

And integers from 1 to 20 for the data. These representations make it easy to write the likelihood function:

class Dice(Suite):
    def Likelihood(self, data, hypo):
        if hypo < data:
            return 0
        else:
            return 1.0/hypo

Here’s how Likelihood works. If hypo<data, that means the roll is greater than the number of sides on the die. That can’t happen, so the likelihood is 0.

Otherwise the question is, “Given that there are hypo sides, what is the chance of rolling data?” The answer is 1/hypo, regardless of data.

Here is the statement that does the update (if I roll a 6):

    suite.Update(6)

And here is the posterior distribution:

4 0.0
6 0.392156862745
8 0.294117647059
12 0.196078431373
20 0.117647058824

After we roll a 6, the probability for the 4-sided die ...