It’s not about the odds; it’s about Beta distribution. You are right to be suspicious about the addition of odds, but there is nothing wrong with adding shape parameters of Beta distributions.
I don’t want to go into many details but a teaser for readers:
One want’s to figure out p, probability of some event happening.
One starts with some prior about p; if it’s uniform over [0,1], the prior is B(1,1).
If one then observes a successes and b failures, one would update to Beta(a+1,b+1).
If one then wants to get the probability of success next time, one needs to integrate over possible p (basically to take expected value). It would lead to ˆp=a+1a+b+2.
For Laplace’s law of succession, you start with B(1,1), observe n failures and update to B(1,n+1). And your estimate is 1n+2. In this context, Pablo suggests starting with different prior of B(1,20) (which corresponds to the probability of success 121 and odds of 1:20) to then update to B(1,90) after observing 70 failures.
It’s not about the odds; it’s about Beta distribution. You are right to be suspicious about the addition of odds, but there is nothing wrong with adding shape parameters of Beta distributions.
I don’t want to go into many details but a teaser for readers:
One want’s to figure out p, probability of some event happening.
One starts with some prior about p; if it’s uniform over [0,1], the prior is B(1,1).
If one then observes a successes and b failures, one would update to Beta(a+1,b+1).
If one then wants to get the probability of success next time, one needs to integrate over possible p (basically to take expected value). It would lead to ˆp=a+1a+b+2.
For Laplace’s law of succession, you start with B(1,1), observe n failures and update to B(1,n+1). And your estimate is 1n+2. In this context, Pablo suggests starting with different prior of B(1,20) (which corresponds to the probability of success 121 and odds of 1:20) to then update to B(1,90) after observing 70 failures.
I can confirm this is correct.
By the way, a similar modelling approach (Beta prior, binomial likelihood function) was used in this report.
This (pointed me to something that) makes so much more sense, thanks Misha, strongly updated.