Bayes Billiards with Tom Crawford

Bayes' Theorem allows us to assign a probability to an unknown fact.

Thomas Bayes himself described an experiment with a billiard table, which is brilliantly explained by Hannah Fry and Matt Parker here    • Bayesian Statistics with Hannah Fry  

Brian Cox and David Spiegelhalter did a 1-dimensional version similar to our experiment here    • People of Science with Brian Cox - Si...  

After we filmed this video, 3blue1brown released his own Bayesian video    • Bayes theorem, the geometry of changi...  

For more of Tom Crawford see his channel    / tomrocksmaths  

Our experiment failed pretty badly really. For some behind-the-scenes information, this was our third attempt at the experiment, the first two were a little slow. The previous attempts were a lot more accurate. Oh well.

Why did we fail? Maybe because the balls were colliding they were not independent. Maybe Tom wasn't random enough. In which case, our assumption that each position is equally likely could be updated.

For more information on this experiment see https://www.nature.com/articles/nbt09...
The main point, I believe, is that for limited data, the Bayesian approach (using the average estimate) is more accurate.

Viewer, Penny Lane, has run a simulation of this experiment, which did show that Bayesian was slightly more accurate than Frequentist (so this real-life attempt was probably a toss up for who did best):
"Here's the output of my script:
simulated 14 balls 10000000 times
the Bayesian approach won over the Frequentist one 50.44934% of the times
6.65762% of the simulations were ties, so 42.89304% were Frequentist wins
the mean deviation from the real value for the Bayesian approach was: 0.0800043179873167
the mean deviation from the real value for the Frequentist approach was: 0.08422584547321381"
See the comment here    • Bayes Billiards with Tom Crawford  
And code here https://pastebin.com/yyCjtnEu

Another comment I liked talked about what would happen if all the balls had been to the right. In that case the frequentist approach would put the position on the extreme left, p=0. While the Bayesian approach would put the position at p = 1/16 = 0.0625, so a little way from the extreme. And that sounds sensible.

People who read the description are the best people. If you have read this, I probably need cheering up after the failure of this experiment, so tell me a joke in the comments. Thanks.