Is the term 'interpretation' in quantum mechanics the same as the term 'interpretation' in probability? For example, the prefix term 'micro-' in 'microfinance' has a different purpose from the 'micro-' in 'microeconomics'.
I heard all 'interpretations' of quantum mechanics
give exactly the same answer to every measurement so they are all equally correct. Is that the same use of the term 'interpretations' as in 'interpretations' of probability?
Context:

*

*In 2011, learned mathematical (frequentist) statistics in 2011 as a quant undergrad.


*In 2014, I encountered the aforementioned.

Sheldon: Okay. Um, what is the correct interpretation of quantum mechanics?


Howard: Since every interpretation gives exactly the same answer to every measurement, they are all equally correct. However, I know you believe in the Many Worlds Interpretation, so I’ll say that. Now do you think I’m smart enough?



*In 2015, I discovered the Bayesian interpretation of probability as a quant grad eg Bayesian logit model - intuitive explanation? and that 99% of my statistics were frequentist.

So Bayesians and frequentists interpret probability differently leading to the things like Lindley's parardox, but they both follow Kolmogorov's axioms and Bayes' theorem so they will never differ on $\mathbb P(A)$ where $A$ is an event in $(\Omega, \mathscr F, \mathbb P)$
Is that the same idea as the use of the term 'interpretation' in quantum mechanics?
If no, why exactly?
If so, elaborate if you want.
 A: There are similarities and differences.
In probability, both interpretations give the same $P(A)$ for an event. But the interpretation of the even is the same: the event is absolute, i.e. it really happened.
In quantum mechanics, all interpretations give the same probabilities for a measurement outcome but the interpretation of the measurement outcome itself may vary.
In collapse theories, the measurement outcome is absolute and really happens, but causality is violated.
In many worlds, there is no single measurement outcome. The world becomes a superposition two worlds, each with its own outcome, and the observer in each world seeing (subjectively) a single outcome.
In Quantum Bayesianism, the probabilities and the measurement outcomes are completely subjective. This is actually a reformulation of probability theory where probabilities can interfere like waves. (This is because, in QM, the probability for a measurement $a$ is given by the square of the probability amplitude, $p=|\psi(a)|^2$. Since $\psi$ is, in general, a complex number, we can get interference between different classical processes by taking superpositions $\psi = \psi_1+\psi_2$.)
A recent paper (referred to in the following article) makes some of these ideas rigorous:
https://www.sciencemag.org/news/2020/08/quantum-paradox-points-shaky-foundations-reality
But the point here is that the different interpretations of QM predict a different final state of the world and of the system after a measurement is made. However, how that measurement outcome is perceived by the observer making the measurement will be the same in all interpretations. So there are no observable differences between the different interpretations, only inobservable differences.
A: Interpretation in quantum mechanics is explaining how seemingly counter-intuitive quantum mechanical behavior comes about. There exist many interpretations.
Interpretations of probability are explaining what probability is (i.e., how it is correctly defined). There are exactly two of them: frequentist and Bayesian interpretations - I do not discuss them here, since I have done it just yesterday in an answer to a different question.
In other words, these are two different disciplines that require explaining/interpreting rather different things. (Moreover, QM is a physical discipline, whereas probability theory is more a mathematical one, although the need for interpretation comes from applying it to real data - not necessarily physics data.)
In terms of philosophical standpoints there is some overlap between the interpretations of QM and those of probability, as pointed out in the answer by @EricDavidKramer.
A: It has been said that where conventional (frequentist) probability looks forward and asks what probably will happen, Bayesian probability looks back and asks what probably caused something to happen.
This can be seen as reflecting time-reversal symmetry in quantum theory, they are not different interpretations of the theory.
Interpretations of quantum physics are concerned with the underlying physical mechanism; does God play dice with us or is there some hidden variable we have not figured out? What goes on in between measurements? Are quantum interactions transactional? Does reality constantly split into multiple parallel universes (yes, really, some physicists take this seriously!)? and so on.
It is probably (sic) correct to say that interpretations of probability play into some interpretations of QM.
A: The non-probabilistic interpretation
The non-probabilistic interpretation of quantum mechanics proposes a mechanism underneath the surface that leads to a probabilistic interpretation of QM on the surface. The Bohmian interpretation is an example: underneath the surface lie hidden variables which lead to a probabilistic theory, but one that is explainable (in contrast to a truly probabilistic theory). Compare it to the unpredictable Brownian motion in a fluid, where the fluid represents the hidden variables (continuous or discrete around very small distances, say the Planck length)
Non-local Hidden Variables interpretations (like Bohmian QM) automatically are non-probabilistic, since an underlying deterministic mechanism is incorporated in this theory/interpretation.
The probabilistic interpretation
This interpretation states that QM is inherent, non-reducible probabilistic. The question remains: Why is this so? There is no answer to this question in this interpretation. It was just stated for a fact (by Born).
Presently, most interpretations are probabilistic in their approach. The question then remains: How can anything be fundamentally probabilistic? The Bohmian interpretation gets rid of this unexplainable interpretation (which was just stated to be probabilistic by Born to explain the outcome of observations; it could, however just as well have been that today we would be talking about a non-probabilistic interpretation, which seems to make more sense to me).
A: Bayesianism often seems to be regarded as a modern interpretation of probability theory, which displaced frequentism, but actually it was the way in which probability theory was always understood except for a short period in the first part of the C20th. I think it was well summarised by Maxwell:

“The actual science of logic is conversant at present only with things either certain,
impossible, or entirely doubtful, none of which (fortunately) we have to reason
on. Therefore the true logic for this world is the calculus of Probabilities,
which takes account of the magnitude of the probability which is, or ought to be,
in a reasonable man’s mind.” James Clerk Maxwell (ed. P. M. Harman), 1990, The Scientific Letters and Papers of James Clerk Maxwell, Vol. 1, 1846-1862, p.197, Cambridge University Press

Frequentism is then seen as a particular way in which probabilities can be assigned in a reasonable man's mind. IOW Bayesianism and Frequentism are not opposing interpretations, but rather Bayesianism is a general interpretation and Frequentism amounts to a special case, which does have a great deal of practical application.
I would contrast this with so-called interpretations of quantum mechanics. These are just waffle. Even von Neumann's quantum logic is so mangled by Wikipedia that it has no bearing on what quantum logic actually is.
The predictions of quantum mechanics are not derived from interpretation, but from the mathematical structure defined by the Dirac–von Neumann axioms. Interpretations, by and large, consist of hand wavy arguments which bear very little relation to the axioms. In some cases, such as Bohmian mechanics, they ignore theorems following from the axioms (which prohibit determinism). In others, such as many worlds, they bear no relation to the axioms, and discuss things like "the wave function of the universe" which is a nonsensical concept in axiomatic quantum mechanics.
Many interpretations, particularly information theoretic interpretions, and interpretations like quantum Bayesianism, and Saul Youssef's "exotic probability theory" simply regurgitate in different words what was already said by Dirac and von Neumann, without adding anything of interest and usually without making direct reference to mathematical structure.
The exception is the orthodox, or Dirac–von Neumann interpretation, in which qm is understood as a mathematical structure for the calculation of probabilities of outcomes of measurement (either Bayesian or frequentist probability can be used). The difference in mathematical structure between this and classical probability theory is precisely that quantum mechanics does not admit hidden variables, whereas in classical probability theory it is often assumed that results are determined by unknowns. The reasons for this mathematical structure are the subject of theorems, not interpretation. I have written much more on this in my books (see profile) and in e.g. The Hilbert space of conditional clauses
