As a non-mathematical person who sometimes uses these tests in biological experiments, I find it frustrating why statisticians can't just give us something we could indeed use with confidence that it actually tells us what we think it tells us...
As someone trained in classic statistical methods who has not had occasion to really use Bayesian methods, one of the things I like about Bayesian approaches is that they seem much more real world and practical to me. E.g. most statistical analysis tools assume a bunch of ridiculous and impossible things (e.g. that you do not look at the data until it has all been collected; that every time you look at the data you need to add a "degree of freedom" meaning that looking at the data 10 times produces 10 increasingly less reliable results -- literally NO-ONE does this and yet it is crucial for the theory to work.)
The Bayesian approach is more like "as you get data, make and refine estimates of what the data means" which is how we actually do things, and so -- everything else aside -- we aren't violating the approach's base assumptions every time we use it.
I'm still not clear on this, what does it mean? I also tried to read the linked text but it's quite dense and drawn out.
Imagine two studies have been completed and they have non-overlapping confidence intervals each of 95%. Does each have a 95% chance of having captured the result? Obviously not -- at most one may be correct.
This is on a slightly different statistical methodology issue, but quoting from Brad Efron (http://statweb.stanford.edu/~ckirby/brad/papers/2005BayesFre...):
The physicists I talked with were really bothered by our 250 year old Bayesian-frequentist argument. Basically there’s only one way of doing physics but there seems to be at least two ways to do statistics, and they don’t always give the same answers.
This says something about the special nature of our field. Most scientists study some aspect of nature, rocks, stars, particles; we study scientists, or at least scientific data. Statistics is an information science, the first and most fully developed information science. Maybe it’s not surprising then that there is more than one way to think about an abstract subject like “information”.