When using Bayesian inference for parameter estimation, one has several choices as to how to present your results. In question #4 on HW2, you are asked to calculate the posterior pdf for f, the binary fraction of O stars, given 3 companion detections in 21 systems. If you were presenting this result in a paper, how would you characterize it? What is the answer?
Well, in Bayesian parameter estimation, the answer is the entire posterior pdf:
But of course, you need to quote a result somewhere in the text, and perhaps in a table, in order that someone else can read your paper and quickly take your result and use it in their work. So what do you do? Do you quote the median with a symmetric 68% confidence interval?
This is the easiest one to calculate, but with an asymmetric pdf like this it doesn't actually look that great, in my opinion. So then do you quote the most likely value, with a different confidence interval? (I believe this one is the "shortest confidence interval.")
Maybe looks a bit nicer, but really, who's to judge? Anyway, the point is that exactly the same data and analysis gives slightly different "results," depending on what you choose to quote.
This is funny, because by summarizing "results" like this you're actually encouraging others to play the "assume a normal distribution with a particular sigma" game. And I fully admit to having done this myself: taking quoted results that I know don't come from a normal posterior distribution and approximating them as normal in my analysis. What can I say, sometimes we're just lazy...who wants to track down a whole bunch of exact posterior distributions from a dozen different authors? But at least this way I can comfort myself by knowing that while I'm taking shortcuts, I know that I'm taking them....