# Week 1 Bayesian Quarterback Rankings

### Adjusting advanced stats for sample size to most accurately project future quarterback efficiency

Everyone seems to be in the quarterback ranking game now, so I might as well throw my hat into the ring. I like to think of my Bayesian quarterback rankings as a check on perceptions, which sometimes are more correct than the cold, hard evidence, but more often not.

I’ll give myself a pat on the back to bringing Bayesian updating to the football analysis game around four years ago, answering the question (more like disproving the assertion) that **Deshaun Watson** should be considered on the same level as **Patrick Mahomes**. While that analysis was a big “hit” for trusting the numbers over recency bias, I have also had some misses, like suggesting that Baker Mayfield wasn’t too far away from Lamar Jackson a couple years into their careers. I’ll take the convenient way out and blame that miss mostly on PFF passing grades that had a significantly more positive view of Mayfield than his advanced metrics, like expected points added (EPA) per play. In this analysis, I’m strictly using historical EPA numbers to project out future quarterback efficiency, since generating points (not grades) is what we really care about.

## QUICK METHODOLOGY PRIMER

If you want the full explanation of Bayesian updating, please refer to this analysis I wrote in the past (which I believe isn’t behind the PFF paywall).

Basically, Bayesian updating begins with an expectation (prior beliefs), then updates the expectation when we get new evidence (posterior beliefs). If the evidence is better than of expectation, the updated expectation is better, and vice-versa. The key is that the evidence only shifts our expectation, it doesn’t define them. The more evidence we get, the more confident that we can be that the evidence is a better representation of actual “truth” than our original expectation. For this analysis, the original expectation is based on historical averages by draft position.

You can see why this would be useful in football analysis, where we’re long on domain knowledge but often short on sample size. In Bayesian statistics, even the smallest sample can be used to update our prior, or initial, forecast. Our prior starts as a strong anchor for our forecast, then waning in influence as our sample of evidence grows larger. This also gives us a range of outcomes, which narrow as we get more evidence. This aligns with common knowledge that quarterback with less experience and play on the field have a larger range of who they will eventually prove to be.

The formula for Bayesian updating requires estimates for “true” range of outcomes (I used the distribution of franchise quarterback efficiency, minimum 1,000 career dropbacks) and standard deviation per piece of evidence (I used the standard deviation of EPA on each quarterback play).

The posterior estimates include a “true” estimate, which is the mean/median of the projected range of outcomes, with a standard deviation assumption that shrinks as quarterbacks get more reps. In the table below, you’ll just find the mean values, but remember that sample size is key, so there are wider distributions, and more quickly updating posteriors, for younger and inexperienced quarterbacks. As an example, here are the distributions for two quarterbacks who look roughly average by mean estimate, but have vastly different samples: **Derek Carr** and **Kenny Pickett**.

It’s very unlikely at this point that Carr is either an elite quarterback, or a poor one, which is reflected in his narrow distribution. Pickett, on the other hand, could still have a large range of outcomes.

## 2023 WEEK 1 PROJECTED EFFICIENCY

These results are the ranking for the go-forward projections of quarterback efficiency this season. I also included the EPA per play rankings for each quarterback over the last six seasons (minimum 250 dropbacks) so you can see the evidence going into the projections.

Older data is decayed over time, so the 2022 EPA per play data matters more than those from pre-2020. That said, older data can’t be fully discounted, or else you miss bounce-back performers of great quarterback returning to form, like **Aaron Rodgers** in 2020 and 2021.

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