Key insight here is you can think of Bayes as requiring:
- Some data
- A generative model
The motivating example was a fish-of-the-month club. You run an experiment where 6 out of 16 people (data) sign up after receiving a brochure. You want to get an estimate of the true sign-up rate. You model the sign up choice as a binomial with some unknown p (generative model) and you don’t have a strong thought on what the actual sign up rate is (uniform prior between 0% and 100%).
We will generate our posterior “guess” of the actual p as follows (Approximate Bayesian Computation):
Pick a p from the prior (uniform so p=.1 and p=.9 are equally likely to be selected)
Plug the selected p into a binomial model, simulate the binomial with n=16 and p=p
If the proportion of “successful” flips in the simulation equals the 6 we observed in the data, count the trial as a success.
Repeat thousands of times.
What you end up with is a posterior distribution of the p’s most likely to generate the 6 out of 16 you saw. This looks normal around 37.5%. This is Bayes in a nutshell. You take your prior, gather some evidence, and update your prior into the posterior to incorporate your evidence.
THOUGHTS: very useful video series to generate some Bayesian intuition.