**Part 1**

Here’s the golf putting data we were using, typed in from Don Berry’s 1996 textbook. The columns are distance in feet from the hole, number of tries, and number of successes:

x n y 2 1443 1346 3 694 577 4 455 337 5 353 208 6 272 149 7 256 136 8 240 111 9 217 69 10 200 67 11 237 75 12 202 52 13 192 46 14 174 54 15 167 28 16 201 27 17 195 31 18 191 33 19 147 20 20 152 24

Graphed here:

Here’s the idealized picture of the golf putt, where the only uncertainty is the angle of the shot:

Which we assume is normally distributed:

And here’s the model expressed in Stan:

data { int J; int n[J]; vector[J] x; int y[J]; real r; real R; } parameters { realsigma; } model { vector[J] p; for (j in 1:J){ p[j] = 2*Phi(asin((R-r)/x[j]) / sigma) - 1; } y ~ binomial(n, p); } generated quantities { real sigma_degrees; sigma_degrees = (180/pi())*sigma; }

Fit to the above data, the estimate of sigma_degrees is 1.5. And here’s the fit:

**Part 2**

The other day, Mark Broadie came to my office and shared a larger dataset, from 2016-2018. I’m assuming the distances are continuous numbers because the putts have exact distance measurements and have been divided into bins by distance, with the numbers below representing the average distance in each bin.

x n y 0.28 45198 45183 0.97 183020 182899 1.93 169503 168594 2.92 113094 108953 3.93 73855 64740 4.94 53659 41106 5.94 42991 28205 6.95 37050 21334 7.95 33275 16615 8.95 30836 13503 9.95 28637 11060 10.95 26239 9032 11.95 24636 7687 12.95 22876 6432 14.43 41267 9813 16.43 35712 7196 18.44 31573 5290 20.44 28280 4086 21.95 13238 1642 24.39 46570 4767 28.40 38422 2980 32.39 31641 1996 36.39 25604 1327 40.37 20366 834 44.38 15977 559 48.37 11770 311 52.36 8708 231 57.25 8878 204 63.23 5492 103 69.18 3087 35 75.19 1742 24

Comparing the two datasets in the range 0-20 feet, the success rate is similar for longer putts but is much higher than before for the short putts. This could be a measurement issue, if the distances to the hole are only approximate for the old data.

Beyond 20 feet, the empirical success rates are lower than would be predicted by the old model. This makes sense: for longer putts, the angle isn’t the only thing you need to control; you also need to get the distance right too.

So Broadie fit a new model in Stan. See here and here for further details.