• Flat priors aren't well explained.
  From @johndcook There are many approaches to this problem. Here are three.

The subjective Bayes approach says the prior should simply quantify what is known or believed before the experiment takes place. Period. End of discussion.

The empirical Bayes approach says you can estimate your prior from the data itself. (In that case your "prior" isn't prior at all.)

The objective Bayes approach says to pick priors based on mathematical properties, such as "reference" priors that in some sense maximize information gain. Jim Berger gives a good defense of objective Bayes here.

In practice someone may use any and all of these approaches, even within the same model. For example, they may use a subjective prior on parameters where there is a considerable amount of prior knowledge and use a reference prior on other parameters that are less important or less understood.

Often it simply doesn't matter much what prior you use. For example, you might show that a variety of priors, say an optimistic prior and a pessimistic prior, lead to essentially the same conclusion. This is particularly the case when there's a lot of data: the impact of the prior fades as data accrue. But for other applications, such as hypothesis testing, priors matter more.

Definition: A prior distribution is non-informative if the prior is “flat”
relative to the likelihood function.

• We chose these priors because in this case we are trying to have as little impact on the posterior distribution as possible. We are trying to have a reasonable model and let inference happen from the data set.

See this https://github.com/albahnsen/CostSensitiveClassification/tree/gh-pages/tutorials

I wonder if something similar is going on for other teams. Let's check.

• See the bottom of http://danielweitzenfeld.github.io/passtheroc/blog/2014/10/28/bayes-premier-league/
  See what this looks like for Rugby in Python.

Add something like
To ensure identifiability, they impose a sum-to-zero constraint on the attack and defense parameters. They also tried a corner constraint (pinning one team's parameters to 0,0), but found that interpretation is easier in the sum-to-zero case because it's not relative to the 0,0 team.

∑t=1Tattt=0
∑t=1Tdeft=0

The hyper-priors on the attack and defense parameters are also flat:

μatt∼Normal(0,.0001)
μdef∼Normal(0,.0001)
τatt∼Gamma(.1,.1)
τdef∼Gamma(.1,.1)

• See this http://danielweitzenfeld.github.io/passtheroc/blog/2014/10/28/bayes-premier-league/

• I think it's a good idea to try again to replicate the model of the teams.
  http://danielweitzenfeld.github.io/passtheroc/blog/2014/10/28/bayes-premier-league/
  Is a good start, I was unable to do this so far, due to some fat fingers. But I think that's a good graph.