I would assume that it’s based on some sort of Bayesian analysis, but I don’t actually know for sure.

It’s a little complicated and technical–and I’m not a Bayesian so I’m probably going to a crappy job explaining it–but I’ll try to summarize what I *think* they’re doing.

First, there are three elements in a Bayesian analysis: prior distribution (original projections), sampling distribution (new data), and posterior distribution (rest of seasons [ROS] projections).

For the sake of simplicity, let’s say that you’re just interested in projecting a Batting Average (a simple binomial has some convenient properties when working within a Bayesian framework). Suppose our prior is .300 but we observe half a season of .250 (75/300). What should the ROS be based on having observed the actual .250 BA over 300 AB compared to an expected .300 based on 3+ years worth of data?

Without getting too far into the weeds, we’d probably estimate what’s called the parameters of a Beta distribution, which yields two shape parameters: alpha and beta. One of the convenient features of a beta distribution is that you can interpret alpha as the number of hits and alpha+beta as the number of at bats. We can then use the BA formula to update the projection.

For sake of simplicity, assume the prior X~Beta(300,1000) and assume 75 hits in 300 AB. Then:

ROS=(300+75)/(1000+300+75)=0.273 (i.e., this is our posterior estimate)

That is, based on having observed 75 hits over 300 at bats (i.e., .250, our sampling distribution) in a half season worth of data, we would make a ROS projection for .273 (our posterior distribution) given those actual data and the prior that he’s a .300 BA hitter (our prior distribution).

Note that I’ve skipped over a few crucial steps to just posit that the prior is X~Beta(300,1000). It’s not as simple as saying a projected .300 hitter implies X~Beta(300,1000). But it keeps the math simple for the purposes of a relatively nontechnical explanation. And you could weaken the prior by assuming/estimating lower values for alpha and beta. If you weaken the strength of the prior, then the posterior will be lower than .273 and closer to a .250 BA. FWIW, one of the reasons why I’m always a little skeptical of Bayesian analysis is that the choice of assumptions about the prior is always a little arbitrary.

Hope that makes some sense. And apologies to any Bayesians who might read this who will be offended by the gross simplifications–it’s a combination of my own limited understanding as well as trying to avoid minimizing the math!

EDIT: Corrected a mistake in my math (I forget to include the alpha=75 in the denominator, so the ROS=.273, not .288 as I originally miscalculated).