"Inverse probability" is a rather old-fashioned way of referring to Bayesian inference; when it's used nowadays it's usually as a nod to history. De Morgan (1838), An Essay on Probabilities, Ch. 3 "On Inverse Probabilities", explains it nicely:
In the preceding chapter, we have calculated the chances of an event,
knowing the circumstances under which it is to happen or fail. We are
now to place ourselves in an inverted position: we know the event, and
ask what is the probability which results from the event in favour of
of any set of circumstances under which the same might have happened.
An example follows using Bayes' Theorem.
I'm not sure that the term mightn't have at some point encompassed putative or proposed non-Bayesian, priorless, methods of getting from $f(y|\theta)$ to $p(\theta|y)$ (in @Christopher Hanck's notation); but at any rate Fisher was clearly distinguishing between "inverse probability" & his methods—maximum likelihood, fiducial inference—by the 1930's. It also strikes me that several early-20th-Century writers seem to view the use of what we now call uninformative/ignorance/reference priors as part & parcel of the "inverse probability" method†, or even of "Bayes' Theorem"‡.
† Fisher (1930), Math. Proc. Camb. Philos. Soc., 26, p 528, "Inverse probability", clearly distinguishes, perhaps for the first time, between Bayesian inference from flat "ignorance" priors ("the inverse argument proper"), the unexceptionable application of Bayes' Theorem when the prior describes aleatory probabilities ("not inverse probability strictly speaking"), & his fiducial argument.
‡ For example, Pearson (1907), Phil. Mag., p365, "On the influence of past experience on future expectation", conflates Bayes' Theorem with the "equal distribution of ignorance".
Best Answer
Yes, your examples are correct.
The Oxford English Dictionary defines pool as:
Another example would be:
you measure blood levels of substance X in males and females. You don't see statistical differences between the two groups so you pool the data together, ignoring the sex of the experimental subject.
Whether it is statistically correct to do so depends very much on the specific case.