![]() I wrote a short Shiny app PredictWinningFraction() to illustrate these calculations. How much do we expect the team to decline? To answer this, we look at the mean of the predictive density of. What is the chance that the team declines (in winning fraction) the following season? To answer this question, we look at the predictive density of the change in winning fractions given, focusing on the probability. Morris shows that this predictive density is normal with mean and standard deviation. So we compute the predictive density of given. One can rewrite the posterior mean asĪctually we are most interested in predicting the team’s fraction in the following season. Morris shows that the posterior mean of the true winning fraction is a compromise between the observed winning fraction and the prior mean. The posterior density for the true winning fraction given is normal( ), where the mean and standard deviation are and. We observe the team’s winning fraction for the first season. Since is known, the Bayesian calculations for this normal sampling/normal prior model are well-known. In this scenario, since teams are playing a 162-game season, a good approximation to the sampling standard deviation for an average team is. The reader can check that the probability that is 0.023, which confirms Morris’ belief that a true winning fraction for the team larger than 0.60 is rare. Also he believes that it is unlikely that the team would win more than 60% of its games in the long run. Morris’ best guess is that this particular division-winning team is average which means that it would win half of its games in the long-run. We assume that is normal with mean and standard deviation. ![]() To complete this model, we put a prior on the team’s ability. Note that we’re assuming the team’s ability does not change from the first to second season. The proportion is the ability of the team - if the team was able to play an infinite number of games, then would represent its long-term winning fraction. ![]() We assume that and are independent where is normal with mean and standard deviation. What is the chance that they will decline in the following season? Let and denote respectively the winning fractions of the team in the two seasons. Suppose a team wins their division in a particular season. Here I describe Morris’ example, use a Shiny app to illustrate the posterior and predictive calculations, and show that the results from this Bayesian model seem to agree with what happens in modern baseball competition. In one of his review papers, Morris provides a nice baseball illustration of a Bayesian model explaining the regression to the mean phenomenon, the idea that extreme performances by teams or individuals tend to move to the average in the following season. In a couple of months, I will giving.a talk at the New England Symposium on Statistics in Sports reviewing Carl Morris’ contributions to statistical thinking in sports. ![]()
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