Posted on 9th April 2019
On my commute, I listen to In Our Time, and I have recently gotten back to the delightful episode on Random and Pseudorandom.
The end of discussion touched upon Kolmogorov Complexity which I wish I knew more about. There was also a discussion about visual randomness, and how humans are terribly bad at judging random arrangements of dots (we think that patterns which are made by inhibiting close points are actually more random than a homogeneous Poisson point process). Peter Coles was name checked in regards to this, and as luck would have it, he has a wonderful blog post all about it.
However, one bit did, yet again, remind me of why I'm a closet Bayesian. To quote:
... apply a test which is called the Chi-squared test, which was invented by the British mathematician Karl Pearson in 1905 and what that does is gives me a way of saying how likely it is, given the proportion of ones, twos and threes that I've seen, that these really did come from chance, and that it really was an unbiased dice that I was throwing.
This is of course not quite correct, because we are really performing a null hypothesis test and so we are really failing to reject the null hypothesis, which in this case is that the dice is not biased. This is made very clear in an wikipedia example. What struck me was that even a trained Mathematician can easily fall into the trap of saying "how likely is it ... came from chance", which is of course a classic mis-interpretation of a p-value.
A Bayesian approach (at the cost of needing a prior, of course) can give an answer to the question "how likely is it that the dice is biased". This has been done to death, and I'll just link to my favourite source, which is David MacKay's book.
This discussion was in the context of randomness tests, about which I'd like to know more, and on which Wikipedia is oddly not particularly detailed.