3 ms·
Surely it isn't that surprising that the outcomes of two different experiments can end up providing the exact same information? In fact this case can be inferr
by contravariant 9y ago
Surely it isn't that surprising that the outcomes of two different experiments can end up providing the exact same information?
In fact this case can be inferred from the fact that the Beta distribution is the conjugate prior distribution for both the binomial and negative binomial distribution. There are many more distributions that have the Beta distribution as conjugate prior to one of the variables (in fact it's pretty much all distributions with a factor p or (1-p) somewhere).
- tel 9y agoIt is often considered surprising and controversial. In particular, if the experiment performed included "controversial" stopping criteria then you could imagine the same data arising but not trusting each result identically. Arguments exist for each side.
- RA_Fisher 9y agoYep, I should add that portion. You're spot on. This was surprising to me because I've shown both models contain the same information but identical Frequentist hypothesis tests will show different p-values. This unshown part is what suprised me. The next post! :)
- contravariant 9y agoSo I guess the problem here is that to calculate the probability that the likelihood exceeds a certain bound can't be calculated solely from the likelihood (+prior) itself. Yet interestingly, the likelihood + prior is enough to make statements about the distribution of the parameters you're looking at. So you can say something about the certainty that your parameter exceeds a certain bound. I suppose it really depends on your application which you'd want to use.
- tel 9y agoThat's a Bayesian perspective, but the heart of the Likelihood Principle is that non-Bayesians/non-Likelihoodists can find reason to believe that the Likelihood doesn't really contain all of the relevant information for making inferences.
- nonbel 9y ago> "all of the relevant information for making inferences" I think this terminology is way too strong. Obviously the context matters. For example, the way the data is collected (whether or not the data collector is a known liar/p-hacker, the sensor is known to malfunction at certain temperatures, etc) should affect inference.
- tel 9y agoRight, and the Likelihoodist/Bayesian would put those details into the model. Real world concerns get handled by each side in acceptable ways, but very special details about how sampling and posterior distributions differ are what make this tricky.
- RA_Fisher 9y agoThanks, I didn't realize it was related to conjugacy!