Allen Downey
This notebook contains a solution to a problem I posed in my Bayesian statistics class:
In the final match of the 2014 FIFA World Cup, Germany defeated Argentina 1-0. How much evidence does this victory provide that Germany had the better team? What is the probability that Germany would win a rematch?
Scoring in games like soccer and hockey can be (reasonably) well modeled by a Poisson process, which assumes that each team, against a given opponent, will score goals at some goal-scoring rate, $\lambda$, and that this rate is stationary; in other words, the probability of scoring a goal is about the same at any point during the game.
Based on this modeling decision, we can answer the questions by
My solution uses the ThinkBayes2 framework, which is described in Think Bayes, and summarized in this notebook.
I'll start with Step 2.
If goal-scoring is a Poisson process, the distribution of goals per game is Poisson with parameter $\lambda$. To compute the distribution of $\lambda$ we can define a new class that inherits from thinkbayes2.Suite and provides an appropriate Likelihood function:
In [1]:
# first a little house-keeping
from __future__ import print_function, division
% matplotlib inline
In [2]:
import thinkbayes2
class Soccer(thinkbayes2.Suite):
"""Represents hypotheses about goal-scoring rates."""
def Likelihood(self, data, hypo):
"""Computes the likelihood of the data under the hypothesis.
hypo: goal rate in goals per game
data: interarrival time in minutes
"""
goals = data
lam = hypo
like = thinkbayes2.EvalPoissonPmf(goals, lam)
return like
Likelihood computes the likelihood of data given hypo, where data is an observed number of goals, and hypo is a hypothetical goal-scoring rate in goals per game. We can compute the likelihood of the data by evaluating the Poisson probability mass function (PMF).
Now we can get back to Step 1.
Before the game starts, what should we believe about each team's goal scoring rate against each other? We could use previous tournament results to construct the priors, but to keep things simple, I'll just use the average goal-scoring rate from all matches in the tournament, which was 2.67 goals per game (total for both teams).
To construct the prior, I'll start with an unrealistic uniform distribution and update it with fake data until the mean matches the observed rate for a single team, 1.34 goals per game.
In [3]:
import numpy
import thinkplot
hypos = numpy.linspace(0, 12, 201)
suite = Soccer(hypos)
suite.Update(0.33) # fake data chosen by trial and error to yield the observed prior mean
thinkplot.Pdf(suite)
suite.Mean()
Out[3]:
In [4]:
germany = suite.Copy(label='Germany')
argentina = suite.Copy(label='Argentina')
germany.Update(1)
argentina.Update(0)
print('posterior mean Germany', germany.Mean())
print('posterior mean Argentina', argentina.Mean())
Update invokes the likelihood function for each hypothetical value of $\lambda$ and updates the distribution accordingly.
Since both teams scored fewer goals than the prior mean (1.4), we expect both posterior means to be lower. Germany's posterior mean is 1.2; Argentina's is 0.7. We can plot the posteriors:
In [5]:
thinkplot.Pdf(germany)
thinkplot.Pdf(argentina)
thinkplot.Config(xlabel='goal-scoring rate', ylabel='probability')
To answer the first question, "How much evidence does this victory provide that Germany had the better team?", we can compute the posterior probability that Germany had a higher goal-scoring rate:
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post_prob = germany > argentina
print('posterior prob Germany > Argentina', post_prob)
Based on the prior distributions, we would have said that Germany had a 50% chance of having the better team, or 1:1 odds. Based on the posteriors, we would say that Germany has a 70% chance. We can use the ratio of the prior and posterior odds to compute the Bayes factor, which measures the strength of the evidence.
In [7]:
prior_odds = 1
post_odds = post_prob / (1 - post_prob)
print('posterior odds Germany > Argentina', post_odds)
k = post_odds / prior_odds
print('Bayes factor', k)
The Bayes factor is 2.4, which is generally considered weak evidence.
Now on to Step 4.
If we knew the actual goal scoring rate, $\lambda$, we could predict how many goals each team work score in a rematch. The distribution of goals would be Poisson with parameter $\lambda t$.
We don't actually know $\lambda$, but we can use the posterior distribution of $\lambda$ to generate a predictive distribution for the number of additional goals.
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def PredictiveDist(suite, duration=1, label='pred'):
"""Computes the distribution of goals scored in a game.
returns: new Pmf (mixture of Poissons)
"""
metapmf = thinkbayes2.Pmf()
for lam, prob in suite.Items():
pred = thinkbayes2.MakePoissonPmf(lam * duration, 15)
metapmf[pred] = prob
mix = thinkbayes2.MakeMixture(metapmf, label=label)
return mix
germany_pred = PredictiveDist(germany, label='germany')
argentina_pred = PredictiveDist(argentina, label='argentina')
PredictiveDist takes the posterior distribution of $\lambda$ and a duration (in units of games).
It loops through the hypotheses in suite, computes the predictive distribution of goals for each hypothesis, and assembles a "meta-Pmf" which is a Pmf that maps from each predictive distribution to its probability.
Finally, it uses MakeMixture to compute the mixture of the distributions. Here's what the predictive distributions look like.
In [9]:
thinkplot.Hist(germany_pred, width=0.45, align='right')
thinkplot.Hist(argentina_pred, width=0.45, align='left')
thinkplot.Config(xlabel='predicted # goals', ylabel='probability', xlim=[-0.5, 7])
Using the predictive distributions, we can compute probabilities for the outcomes of a rematch.
In [10]:
win = germany_pred > argentina_pred
lose = germany_pred < argentina_pred
tie = 1 - (win + lose)
print('posterior prob Germany wins rematch', win)
print('posterior prob Argentina wins rematch', lose)
print('posterior prob tie', tie)
If the score is tied after 90 minutes the teams play an additional 30 minute period. We can use PredictiveDist again to compute the distribution of scores after 1/3 of a game:
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germany_pred_overtime = PredictiveDist(germany, 1/3, label='germany')
argentina_pred_overtime = PredictiveDist(argentina, 1/3, label='argentina')
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win = germany_pred_overtime > argentina_pred_overtime
lose = germany_pred_overtime < argentina_pred_overtime
tie = 1 - (win + lose)
print('posterior prob Germany wins rematch', win)
print('posterior prob Argentina wins rematch', lose)
print('posterior prob tie', tie)
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print('Total prob Germany wins a rematch', 0.45 + 0.32 * 0.26)
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print('Total prob Argentina wins', 0.23 + 0.32 * 0.14)
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print('Prob of draw after overtime', 0.32 * 0.60)
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