world_cup01.ipynb

Kernel: Python 2

Bayesian Statistics Made Simple

Code and exercises from my workshop on Bayesian statistics in Python.

Copyright 2016 Allen Downey

MIT License: https://opensource.org/licenses/MIT

from __future__ import print_function, division

%matplotlib inline

import warnings
warnings.filterwarnings('ignore')

import math
import numpy as np
from scipy.special import gamma

from thinkbayes2 import Pmf, Suite
import thinkplot

The World Cup Problem

We'll use λ to represent the hypothetical goal-scoring rate in goals per game.

To compute prior probabilities for values of λ, I'll use a Gamma distribution.

The mean is 1.3, which is the average number of goals per team per game in World Cup play.

from thinkbayes2 import MakeGammaPmf

xs = np.linspace(0, 8, 101)
pmf = MakeGammaPmf(xs, 1.3)
thinkplot.Pdf(pmf)
thinkplot.Config(xlabel='Goals per game')
pmf.Mean()

1.3103599490022562

Exercise: Write a class called Soccer that extends Suite and defines Likelihood, which should compute the probability of the data (the time between goals in minutes) for a hypothetical goal-scoring rate, lam, in goals per game.

Hint: For a given value of lam, the time between goals is distributed exponentially.

Here's an outline to get you started:

class Soccer(Suite):
    """Represents hypotheses about goal-scoring rates."""

    def Likelihood(self, data, hypo):
        """Computes the likelihood of the data under the hypothesis.

        hypo: scoring rate in goals per game
        data: interarrival time in minutes
        """
        return 1

# Solution goes here

Now we can create a Soccer object and initialize it with the prior Pmf:

soccer = Soccer(pmf)
thinkplot.Pdf(soccer)
thinkplot.Config(xlabel='Goals per game')
soccer.Mean()

1.3103599490022564

Here's the update after first goal at 11 minutes.

thinkplot.Pdf(soccer, color='0.7')
soccer.Update(11)
thinkplot.Pdf(soccer)
thinkplot.Config(xlabel='Goals per game')
soccer.Mean()

1.3103599490022566

Here's the update after the second goal at 23 minutes (the time between first and second goals is 12 minutes).

thinkplot.Pdf(soccer, color='0.7')
soccer.Update(12)
thinkplot.Pdf(soccer)
thinkplot.Config(xlabel='Goals per game')
soccer.Mean()

1.3103599490022566

This distribution represents our belief about lam after two goals.

Estimating the predictive distribution

Now to predict the number of goals in the remaining 67 minutes. There are two sources of uncertainty:

1. We don't know the true value of λ.

2. Even if we did we wouldn't know how many goals would be scored.

We can quantify both sources of uncertainty at the same time, like this:

1. Choose a random values from the posterior distribution of λ.

2. Use the chosen value to generate a random number of goals.

If we run these steps many times, we can estimate the distribution of goals scored.

We can sample a value from the posterior like this:

lam = soccer.Random()
lam

1.04

Given lam, the number of goals scored in the remaining 67 minutes comes from the Poisson distribution with parameter lam * t, with t in units of goals.

So we can generate a random value like this:

t = 67 / 90
np.random.poisson(lam * t)

0

If we generate a large sample, we can see the shape of the distribution:

sample = np.random.poisson(lam * t, size=10000)
pmf = Pmf(sample)
thinkplot.Hist(pmf)
thinkplot.Config(xlabel='Goals scored', ylabel='PMF', xlim=[-0.6, 10.5])
pmf.Mean()

0.777

But that's based on a single value of lam, so it doesn't take into account both sources of uncertainty. Instead, we should sample value values from the posterior distribution and generate one prediction for each.

Exercise: Write a few lines of code to

1. Use Pmf.Sample to generate a sample with n=10000 from the posterior distribution soccer.

2. Use np.random.poisson to generate a random number of goals from the Poisson distribution with parameter $\lambda t$, where t is the remaining time in the game (in units of games).

3. Plot the distribution of the predicted number of goals, and print its mean.

4. What is the probability of scoring 5 or more goals in the remainder of the game?

# Solution goes here

# Solution goes here

Computing the predictive distribution

Alternatively, we can compute the predictive distribution by making a mixture of Poisson distributions.

MakePoissonPmf makes a Pmf that represents a Poisson distribution.

from thinkbayes2 import MakePoissonPmf

If we assume that lam is the mean of the posterior, we can generate a predictive distribution for the number of goals in the remainder of the game.

lam = soccer.Mean()
rem_time = 90 - 23
lt = lam * rem_time / 90
pred = MakePoissonPmf(lt, 10)
thinkplot.Hist(pred)
thinkplot.Config(title='Option 1', 
                 xlabel='Expected goals',
                 xlim=[-0.5, 10.5])

The predictive mean is about 2 goals.

pred.Mean()

0.97549010518228874

And the chance of scoring 5 more goals is still small.

pred.ProbGreater(4)

0.0032977689768040348

But that answer is only approximate because it does not take into account our uncertainty about lam.

The correct method is to compute a weighted mixture of Poisson distributions, one for each possible value of lam.

The following figure shows the different predictive distributions for the different values of lam.

for lam, prob in soccer.Items():
    lt = lam * rem_time / 90
    pred = MakePoissonPmf(lt, 14)
    thinkplot.Pdf(pred, color='gray', alpha=0.3, linewidth=0.5)

thinkplot.Config(xlabel='Expected goals')

We can compute the mixture of these distributions by making a Meta-Pmf that maps from each Poisson Pmf to its probability.

metapmf = Pmf()

for lam, prob in soccer.Items():
    lt = lam * rem_time / 90
    pred = MakePoissonPmf(lt, 15)
    metapmf[pred] = prob

MakeMixture takes a Meta-Pmf (a Pmf that contains Pmfs) and returns a single Pmf that represents the weighted mixture of distributions:

def MakeMixture(metapmf, label='mix'):
    """Make a mixture distribution.

    Args:
      metapmf: Pmf that maps from Pmfs to probs.
      label: string label for the new Pmf.

    Returns: Pmf object.
    """
    mix = Pmf(label=label)
    for pmf, p1 in metapmf.Items():
        for x, p2 in pmf.Items():
            mix[x] += p1 * p2
    return mix

Here's the result for the World Cup problem.

mix = MakeMixture(metapmf)

And here's what the mixture looks like.

thinkplot.Hist(mix)
thinkplot.Config(title='Option 2', 
                 xlabel='Expected goals',
                 xlim=[-0.5, 10.5])

Exercise: Compute the predictive mean and the probability of scoring 5 or more additional goals.

# Solution goes here