Freelance Friday: Bayesian Analysis of 3PT Shooting

Freelance Friday is a regular feature at The Nylon Calculus where we solicit contributions from readers. This edition is a discussion of partial season three point shooting percentages through the lens of Bayes’ Theorem from Nick Neuteufel. Nick is a political scientist who alternates between thinking about public opinion’s impact on foreign policy, especially in electoral and legislative contexts, and NBA analytics. You can follow him @Neuteufel if you want a healthy mix of political analysis and basketball tweets. Questions, comments or submissions for Freelance Friday should be directed to@NylonCalculus on twitter or via email to TheNylonCalculus at Gmail dot com.

Cracking Sample Size: Introducing Bayesian Analysis of 3PT Shooting

Remember five games into the 2015-2016 NBA season? Reigning MVP Steph Curry has gone 52% from behind the arc while MVP runner-up James Harden is shooting a nice 16%. But even trolls on Twitter (actually, especially trolls on Twitter) will tell you that five NBA games don’t mean much. Injuries, scheduling, or even pure dumb luck can sway outcomes or statistics in small sample sizes. Darryl Blackport demonstrated earlier on this site that it takes roughly 750 three-point attempts for three-point percentage to stabilize.

So how do you predict an NBA shooter’s 3-point success over the course of the season given a small sample size way below the 750-attempts threshold? When is it enough to alter your prediction or to hear alarm bells?

Bayesian thinking (named after English minister and statistician Thomas Bayes) provides the answer to those questions. Such thinking begins with a prior belief: before looking at any evidence, what would you guess an NBA shooter’s 3-point percentage would be? Then you look at your new evidence (this season’s stats), weigh the evidence, and shift your belief accordingly. This is called your posterior belief.

Bayesian analysis is a thinking process that is easy to do once you digest it, but putting numbers to it can be difficult at time. When is the new evidence good enough to shift your belief? It varies from person to person, but I present a model here that quantifies a simple Bayesian approach to 3-point analysis[1. Stats teachers and mathematics nerds, please forgive my attempt to balance mathematical rigor with accessible writing for all basketball fans.]. It succinctly answers these questions and tells you when to freak out–and, more importantly, when not to.

Baseball sabermetricians have paved the way for Bayesian analysis of sports percentages. However, baseball analysts have it a bit easy. Baseball has more games and more standardized procedures of play. The vast majority of at-bats and plate appearances are just the same thing repeated with different pitchers, pitch types, and hitters. That’s not the case with NBA shooting. Fewer games, different defensive styles and defenders, varying shooters, and multiple 3-point shot types all make a single analysis of 3PT% impractical or misleading.

The graph below demonstrates this problem. The distribution of 3-point shooting for qualified shooters massively differs based on whether it’s a pull-up 3 attempt (blue region) or a catch & shoot 3 attempt (orange)[2. Data are all available player-seasons data from SportVU for the 2013-14 and 2014-15 seasons qualified for leaderboards.]:

The solution to our 3-point prediction problem is to estimate pull-up and catch & shoot 3-point shooting separately, using different priors and evidence to consider.

What I’ve done is used the data from before this current season to empirically estimate an expected distribution of 3-point success for both pull-up and catch & shoot shooting. These priors are the curves below. The pull-up prior (the blue line) fits the distribution (the blue shaded region) pretty well, while the catch & shoot (orange) fit is good, but not great. This should be fixed as more SportVU data become available over time[2. Important stats note, though if you don’t like math, I’d consider skipping this paragraph. For purposes of fitting the beta priors, I used “qualified shooting,” eliminating 3-point player-seasons below certain thresholds depending on the season because beta distributions are hard to fit with 0% and 100% values. So the best way to interpret these statistics is by thinking of these projections as “considering NBA regular 3-point shooters,” not “considering all NBA players.”]:

Now we have our prior beliefs as to how 3-point success should be distributed by these two shot types. (They should be distributed roughly under the lines in the graph above.) Our prior evidence tells us that the average catch & shoot shooter will have a higher percentage made and that the distribution of pull-up success is more dispersed overall.

So now that we have our priors, we consider and weigh the evidence we have from this season. Through five games, it should have been really hard (if not impossible) to shift our priors that far. Harden shooting terribly in five games should not convince you that he’s a way-below average 3-point shooter–even if you don’t know anything about him before this season. Your prior belief is pretty strong that those games are probably just a cold streak or small noise in the overall scheme of things. However, if that trend continued, that would be a different story. So how do we account for that mathematically?

I’ll yadda-yadda the formal mathematics[3. Ask me on Twitter @Neuteufel for more detail.], but basically one weighs how many attempts and successes (made 3s of each type) and creates a posterior distribution for the player’s expected shooting. The more made shots in good volume, the more we can shift the posterior distribution to the right–we have more evidence of great 3-shooting ability.

Here’s Steph’s new distribution based on data through Dec. 16, 2015:

This is incredible, to put it simply. Even considering potential regression towards the mean, given the volume of evidence we’ve seen of 2015-16 Steph Curry, he is really incredible shooting at an incredible level.

Think of the dashed curves as the new distributions for Curry himself. Given the evidence of his 3-point success this season, this is the credible range of 2015-16 Curry’s 3-point shooting (either off the dribble or on the catch). The peaks are what I call the “Bayes 3PT%” (for math types, the mean of the posterior).

For Curry, these are eye-popping numbers. His Bayes pull-up 3PT% (the peak of the dashed blue curve) is higher than the average NBA-level shooter shoots in catch & shoot situations!  His Bayes catch & shoot 3PT% of 46.9% (the peak of the orange dashed curve) is way above the rest of the league’s shooters’ catch & shoot abilities.

Bayes 3PT% can also be saddening. The more misses the player gives us, the more evidence exists that his inherent shooting ability of a player to shoot his selected shots is below the league average. Here’s 2015-16 Kobe after Dec. 16’s games. It’s not looking good:

2015-16 Kobe is fourth from last in the league’s shooters leaderboard in both Bayes catch & shoot and Bayes pull-up three-point percentage. Byron Scott should consider other long-range options. Bayesian alarm bells have been going off for a while.

If you wish for more of this, here is a link to a Dropbox folder with Bayes 3PT% leaderboard reports. (Note that this is a leaderboard of qualified 3-point shooters, so don’t expect a Hassan Whiteside projection any time soon.)