Last time I’ve argued that Bayes’ technique of learning by updating knowledge, however ideal it is, is not the approach most of us would follow. This time we will see a Bayesian approach that we do, albeit subconsciously, in judging performances. For that, we take the example of Baseball.
Did Jose Iglesias appear to beat the all-time MLB record, which was more than a century-old, when he started the 2013 season with nine hits in 20 bats? His batting average in April was 0.45, with another 330 more batting to go! To most people who knew the historical averages, Jose’s performance might have appeared as a beginner’s luck!
Hierarchical model
One way to analyse Jose’s is using the technique we have used in the past, also known as frequentist statistics. By calculating the mean at the end of April, standard deviation and confidence interval. But we can do better using the historical average as prior data and following the Bayesian approach. Such techniques are known as hierarchical models.
The hierarchical models get the name because the calculations take multiple steps to reach the final estimate. The first level is player to player variability, and the second is the game to game variability of a player.
What we need to predict using the hierarchical model is Jose’s batting average at the end of the season, given that he has hit 45% in the first 20. By then, Jose’s average would be a dot on a larger distribution, the probability distribution of a parameter p (a player’s success probability for this season, but we don’t know that yet), and we assume a normal distribution. We will take the expected value of p or the average to be 0.255, last season’s average, with a standard error (SE) of 0.023 (there is a formula to calculate SE from p). By the way, SE = standard deviation / (square root of N).
Taking beginner’s luck seriously
Jose batted the first 20 at an average of 0.45, and we estimate the standard error of 0.111, as we do for any other probability distribution. If the MLB places its player averages on a normal distribution, Jose today is at the extreme right on 0.45, or an observed average of Y. Its expected value is 0.255!
In our shorthand notation, Y|p ~ N(p, 0.111); we don’t know what p is, but it is more like the probability of success of a Bernoulli trial.
Calculate Posterior Distribution
The objective is to estimate E(p), given that the player has an average Y and standard error SE. The notation is E(p|Y). We express this as a weighted average of all players and Jose. E(p|Y) = B x 0.255 + (1-B) x 0.45, where B is a weightage factor calculated based on the standard errors of the player and the system. B = (0.111)2 /[(0.111)2 + (0.023)2]. As per this, B -> 1 when the player standard error is large and B -> 0 if it is small. In our case B = 0.96. It is not surprising if you look at the standard error of Jose’s performance, which is worse than the overall historical average, simply because of the smaller number (20) of matches he played in 2013 compared to all players in the previous season.
So E(p|Y=0.45) = 0.96 x 0.255 + (1-0.96) x 0.45 = 0.263. This is the updated (posterior) average of Jose.
Jose Iglesias in 2013
MLB Batting Leaders
MLB Batting Averages since 1871
