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Probability, Decisions and Games pdf
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My solution is in this Python file. The problem statement indicates that the prior distribution of p is uniform from 0 to 1. Given a hypothetical value of p and the observed number of wins and losses, we can compute the likelihood of the data under each hypothesis: Billiards inherits the Update function from Suite which is defined in thinkbayes. I left out the first term, the binomial coefficient, because it doesn't depend on p , so it would just get normalized away.
Now to compute the probability that Bob wins the match. Since Alice is ahead 5 points to 3, Bob needs to win the next three points. We don't know the value of p , but we have its posterior distribution, which we can "integrate over" like this:. Using a frequentist approach, we get a substantially different answer. Instead of a posterior distribution, we get a single point estimate. But let's consider why the frequentist result is wrong.
The problem is not the estimate itself. The difference is that Bayesian posterior contains all of the information we have about p , whereas the frequentist result discards a large part of that information. The result we are interested in, the probability of winning the match, is a non-linear transform of p, and in general for a non-linear transform f , the expectation E[ f p ] does not equal f E[ p ].
The Bayesian method computes the first, which is right; the frequentist method approximates the second, which is wrong. To summarize, Bayesian methods are better not just because the results are correct, but more importantly because the results are in a form, the posterior distribution, that lends itself to answering questions and guiding decision-making under uncertainty.
I never said anything about Bayesians and frequentists as people, or about what methods a hypothetical statistician might choose. Any method that produces a point estimate is going to have the same problem. I would rather have a method that is always right than a method that sometimes works and sometimes fails, and requires substantial expertise to know when to expect which.
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