His 1996 Probability in Quantum Theory says:
For some sixty years it has appeared to many physicists that probability plays a fundamentally different role in quantum theory than it does in statistical mechanics and analysis of measurement errors. It is a commonly heard statement that probabilities calculated within a pure state have a different character than the probabilities with which different pure states appear in a mixture, or density matrix. As Pauli put it, the former represents "Eine prinzipielle Unbestimmtheit, nicht nur Unbekanntheit". But this viewpoint leads to so many paradoxes and mysteries that we explore the consequences of the unified view, that all probability signifies only incomplete human information. We examine in detail only one of the issues this raises: the reality of zero point energy.The German means "A fundamental uncertainty, not only obscurity", but probably sounds better when Pauli says it.
A lot of other smart physicists have said that quantum mechanics shows that nature is intrinsically probabilistic, such as R.P. Feynman.
These physicists are smarter than I am, but I say that they are wrong about this. It is crazy to say that probability is a physically real thing, and quantum mechanics does not require such a view. I detailed my opinion in this post on probability last year, and in many other postings and essays on this blog.
Probability is mathematics, not physics. It is essential to nearly all empirical quantitative science, because it gives the tools for comparing theory to experiment. As Jaynes explains, probability gets used in statistical mechanics and analysis of measurement errors, and the use in quantum mechanics is not much different.
Pauli's phrase is reasonable if interpreted to just mean that quantum uncertainty is not just the obscurity of not knowing the values of hidden variables. The hidden variable theories have all failed, as best demonstrated by the Bell test experiments.
Most physicists go farther, and argue that quantum uncertainty is some sort of physical thing that is fundamental to the theory, that proves indeterminism, that can be used for super-Turing computation, and that is even conserved in black holes.
Probability does none of those things. (The world may be indeterministic, but for other reasons.) It is just a mathematical construct that tells us what to expect.
Just look at the picture of Jaynes, and compare it to the wild-haired physicists! Who is more likely to give you the straight truth about the nature of reality?
Here is a statistician squirming about a blown prediction:
Data guru Nate Silver of FiveThirtyEight.com tells NPR's Scott Simon how all the forecasts, including his own, were so far off in predicting the results of this week's British election. ...Probability is what allows him to explain away his failure.
SIMON: Yeah. I've got to tell you, Mr. Silver, you're not giving people much of an incentive to read FiveThirtyEight seriously if you're essentially backing away from the idea that you can reach any conclusions. That's why people read you.
SILVER: Sometimes the right conclusion is to say that people are too sure of themselves, right?
SILVER: Sometimes it's to anticipate uncertainties in your environment. And over the whole course of our life span at FiveThirtyEight - and we've predicted many things in politics and sports and other events for many, many years - historically, one thing that's distinguished us is that our probabilities have been right over the long run. That when we say something has an 80 percent of happening, it happens about 80 percent of the time and it doesn't happen about 20 percent of the time.
Quantum mechanics is similar. If an atom is in a superposition making one energy value 80% likely, then probability is what lets you explain away a measurement that gives another energy value.
The election is analogous to the atom energy measurement. Silver's view is that elections have fundamental uncertainties, so he can just give probabilities. Just like the quantum mechanics professors.