December 1, 2008
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Bayes’ Theorem and prospect valuationby Victor WangAugust 14, 2008 Bayes’ Theorem, named after Thomas Bayes, is a way to determine posterior probabilities after being given a set of prior and conditional probabilities. It has been used before in past baseball analysis. As shown in the linked article, it is useful to use Bayes’ Theorem to “update” a player’s projection since we have prior information, the player’s preseason projection, and we have the conditional information of the plate appearances that a player has accumulated during the season. Most projection systems take a weighted average of a player’s past stats adjusted for context and regress to the mean, and then make an age adjustment. When we regress to the mean, we are assuming that the distribution of player talent is normally distributed. However, if talent is not distributed normally, then regressing to the mean is incorrect and we would want to use Bayesian analysis instead. Coming up with the actual talent distribution for major leaguers is tricky but some research has shown that major league talent approximates a normal distribution. While major league talent may be close to a normal distribution, we know that minor league talent isn’t normally distributed when it comes to future major league production. My research has shown that most top 100 prospects eventually become either supporting players or busts, while fewer become everyday players and even fewer become stars. We can use this prior information of how top prospects perform with Bayes’ Theorem and minor league statistics to come up with individual player values. One of the complaints I have gotten when using my prospect value rankings is that the system is too macro and it needs to incorporate more individual prospect information. Well, we can do this using prior probabilities (prospect group distributions) and conditional information (a player’s minor league equivalency (MLE)). Here is how we can do this:
Here is an example of this process using Andy LaRoche and his statistics coming into 2008:
So given LaRoche’s minor league track record, he has a high probability of becoming an everyday player but a low chance of being a star. While this type of prospect might not seem too valuable, an everyday player cost controlled for six years is immensely valuable as Laroche’s surplus value using this Bayesian analysis coming into 2008 was $40 million. Making some basic assumptions about what Laroche could be expected to be paid, PECOTA had Laroche worth around $50 million in surplus value coming into 2008. So it’s good to see that the Bayesian system has a similar rating with PECOTA. Despite this, there are still some weaknesses with this model:
I hope to keep building on this type of model for prospect valuation. Right now I would definitely recommend using the basic prospect valuations based on a prospect’s ranking for prospects below Double-A. I would also love to hear comments on people’s thoughts of this kind of model and suggestions for improvements. Victor Wang's work on OPS has been featured in SABR's By the Numbers magazine, and was the 2007 recipient of SABR's Jack Kavanagh Memorial Youth Baseball Research Award. He can be reached via email here. Do you have a general question or comment for one of THT's writers? Send it in to our weekly mailbag We also welcome unsolicited op-ed pieces of approximately 500 words for consideration. We reserve the right to edit for length, clarity and consistency of style. Please include your whole name and location to be considered. If you have a comment about this specific article, please email the writer. Next Article: The remains of the season: Los Angeles Dodgers>> <<Previous Article: THT Daily: Astros rocketing |