My current understanding of the notion "confidence interval with confidence level $1 - \alpha$" is that if we tried to calculate the confidence interval many times (each time with a fresh sample), it would contain the correct parameter $1 - \alpha$ of the time.
Though I realize that this is not the same as "probability that the true parameter lies in this interval", there's something I want to clarify.
Before we calculate a 95% confidence interval, there is a 95% probability that the interval we calculate will cover the true parameter. After we've calculated the confidence interval and obtained a particular interval $[a,b]$, we can no longer say this. We can't even make some sort of non-frequentist argument that we're 95% sure the true parameter will lie in $[a,b]$; for if we could, it would contradict counterexamples such as this one: What, precisely, is a confidence interval?
I don't want to make this a debate about the philosophy of probability; instead, I'm looking for a precise, mathematical explanation of the how and why seeing the particular interval $[a,b]$ changes (or doesn't change) the 95% probability we had before seeing that interval. If you argue that "after seeing the interval, the notion of probability no longer makes sense", then fine, let's work in an interpretation of probability in which it does make sense.
Suppose we program a computer to calculate a 95% confidence interval. The computer does some number crunching, calculates an interval, and refuses to show me the interval until I enter a password. Before I've entered the password and seen the interval (but after the computer has already calculated it), what's the probability that the interval will contain the true parameter? It's 95%, and this part is not up for debate: this is the interpretation of probability that I'm interested in for this particular question (I realize there are major philosophical issues that I'm suppressing, and this is intentional).
But as soon as I type in the password and make the computer show me the interval it calculated, the probability (that the interval contains the true parameter) could change. Any claim that this probability never changes would contradict the counterexample above. In this counterexample, the probability could change from 50% to 100%, but...
Are there any examples where the probability changes to something other than 100% or 0% (EDIT: and if so, what are they)?
Are there any examples where the probability doesn't change after seeing the particular interval $[a,b]$ (i.e. the probability that the true parameter lies in $[a,b]$ is still 95%)?
How (and why) does the probability change in general after seeing the computer spit out $[a,b]$?
Thanks for all the great answers and helpful discussions!