book

Think Bayes

by Allen B. Downey

September 2013

Beginner

210 pages

4h 38m

English

O'Reilly Media, Inc.

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My theory, which is mineModeling and approximationWorking with the codeCode stylePrerequisitesConventions Used in This BookSafari® Books OnlineHow to Contact UsContributor List
Conditional probabilityConjoint probabilityThe cookie problemBayes’s theoremThe diachronic interpretationThe M&M problemThe Monty Hall problemDiscussion
DistributionsThe cookie problemThe Bayesian frameworkThe Monty Hall problemEncapsulating the frameworkThe M&M problemDiscussionExercises
The dice problemThe locomotive problemWhat about that prior?An alternative priorCredible intervalsCumulative distribution functionsThe German tank problemDiscussionExercises
The Euro problemSummarizing the posteriorSwamping the priorsOptimizationThe beta distributionDiscussionExercises
OddsThe odds form of Bayes’s theoremOliver’s bloodAddendsMaximaMixturesDiscussion
The Price is Right problemThe priorProbability density functionsRepresenting PDFsModeling the contestantsLikelihoodUpdateOptimal biddingDiscussion
The Boston Bruins problemPoisson processesThe posteriorsThe distribution of goalsThe probability of winningSudden deathDiscussionExercises
The Red Line problemThe modelWait timesPredicting wait timesEstimating the arrival rateIncorporating uncertaintyDecision analysisDiscussionExercises
PaintballThe suiteTrigonometryLikelihoodJoint distributionsConditional distributionsCredible intervalsDiscussionExercises

The Variability HypothesisMean and standard deviationUpdateThe posterior distribution of CVUnderflowLog-likelihoodA little optimizationABCRobust estimationWho is more variable?DiscussionExercises
Back to the Euro problemMaking a fair comparisonThe triangle priorDiscussionExercises
Interpreting SAT scoresThe scaleThe priorPosteriorA better modelCalibrationPosterior distribution of efficacyPredictive distributionDiscussion
The Kidney Tumor problemA simple modelA more general modelImplementationCaching the joint distributionConditional distributionsSerial CorrelationDiscussion
The Geiger counter problemStart simpleMake it hierarchicalA little optimizationExtracting the posteriorsDiscussionExercises
Belly button bacteriaLions and tigers and bearsThe hierarchical versionRandom samplingOptimizationCollapsing the hierarchyOne more problemWe’re not done yetThe belly button dataPredictive distributionsJoint posteriorCoverageDiscussion

Content preview from Think Bayes

Chapter 4. More Estimation

The Euro problem

In Information Theory, Inference, and Learning Algorithms, David MacKay poses this problem:

A statistical statement appeared in “The Guardian” on Friday January 4, 2002:
When spun on edge 250 times, a Belgian one-euro coin came up heads 140 times and tails 110. ‘It looks very suspicious to me,’ said Barry Blight, a statistics lecturer at the London School of Economics. ‘If the coin were unbiased, the chance of getting a result as extreme as that would be less than 7%.’
But do these data give evidence that the coin is biased rather than fair?

To answer that question, we’ll proceed in two steps. The first is to estimate the probability that the coin lands face up. The second is to evaluate whether the data support the hypothesis that the coin is biased.

You can download the code in this section from http://thinkbayes.com/euro.py. For more information see “Working with the code”.

Any given coin has some probability, x, of landing heads up when spun on edge. It seems reasonable to believe that the value of x depends on some physical characteristics of the coin, primarily the distribution of weight.

If a coin is perfectly balanced, we expect x to be close to 50%, but for a lopsided coin, x might be substantially different. We can use Bayes’s theorem and the observed data to estimate x.

Let’s define 101 hypotheses, where H_x is the hypothesis that the probability of heads is x%, for values from 0 to 100. I’ll start with a uniform prior where the probability ...