But if we do talk about that though. Yes, science proceeds by modifying or sometimes totally abandoning the assumptions. No silver bullet, and the point is to assume as little as possible.
In general, if things "just work", its not a reason to abandon attempts to understand them. Things "just work" until they don't, especially in finance.
Consider two guys. Guy A, he predicts that the sun rises every day and is there until dusk. It "works", and so, as the joke goes, he does not touch anything. And guy B, he knows physics and has telescopes. And he can predict the solar eclipses. He knows the failure modes of the model of guy A. Consider the difference between the two.
If all that astronomy offered was a more verbose description of why the sun rises, yielding literally zero different predictions from Guy A’s super simple model, no one would care, and might call Guy B a witch!
I’d argue that what matters for science is pure predictive efficacy. That’s it. If you can explain something, it means you can accurately predict something about an unknown state of affairs that would falsify your model if your prediction is wrong. That’s it. Any other kind of explainability is just a matter of linguistic convenience.
Of course there is overfitting, etc., but that’s just part of refining the model to yield greater predictive accuracy on unseen data (less generalization error). It’s still all about putting your predictions where your mouth is.
If the differing theories of Guy A and Guy B cannot be separated by actually testing the predictions they make, then there is no “interpretability” — just linguistic hand waving.
Incidentally I think a lot of modern focus on interpretability is actually about how to be a political gatekeeper or taste-maker through linguistic hand waving, and is not about developing models who better survive the rigors of being required to make real predictions about states of affairs.
How does this invalidate the notion of interpretability?
A model can be both wrong/misapplied and fully interpretable.
Who said that it did?
I’m saying the development of a model that yields scientific progress doesn’t have to have a connection to the parent comment’s proposed definition of interpretability, and may have competing interpretability concerns driven by specific inference goals that have nothing to do with proofs about the model under assumptions.
Let's say someone used a neural network to predict planetary positions as a function of time, mass and so on. They made a network with several hundred parameters, trained until it stopped converging and published their final state vector.
Would that ever lead to gravitational theory?
Another point of view would be to say how does the Wiles proof of Fermat’s Last Theorem give us “understanding” (or substitute virtually any non-constructive existence proof, or proofs by contradiction, such as for the Halting Problem).
Compact math descriptions are good, don’t get me wrong. When the mechanism of some data generating process is actually oriented in such a way that we can get a concise math description then we absolutely should. But that is largely unrelated to advancing understanding or scientific progress, which can perfectly be some ugly enumerated garble of parameters, or some extremely piecewise solution space. Not every data generating process is required to admit a concise mathematical description.
F = G (m1 * m2) / r^2
Newton's theory of gravitation identified the relevant variables and their relationships. It had no junk variables or parameters. You can do algebraic transformations on it and use the formula outside of the original context. It was not fully correct, but it made exact predictions. People were able to notice the discrepancies it had with orbits of some celestial bodies, which helped in identifying general theory of relativity.
All of this would have been lost of the formula was a state vector of a large neural network.
No, it introduced a useful low-dimensional fiction that approximately works, with high accuracy at coarse scales, and doesn’t account for relativistic or quantum effects (and so, in an absolute sense, is only an approximation, not an interpretation of what is real).
Besides, what you glibly type out as a few characters of a formula unpackages into a deeply hard-to-explain theory in natural language. It’s vastly less interpretable than simpler (and less correct) models because it introduces much more complexity of the meaning and physical concepts behind the formula terms. Just as quantum or relativistic extensions add even more complexity and counter-intuitive concepts and become even harder to interpret.
> “All of this would have been lost of the formula was a state vector of a large neural network.”
Why? I see no reason why that would be true. If there is a very low-dimensional manifold in parameter space that accounts for all the predictive power, that would be a very easy thing to diagnose from a neural network model. Lots of things have been discovered that way, like graph centrality metrics coming from eigen decompositions, style transfer, glove vectors, eigenfaces.
Getting the high-dimensional representation that yields accurate predictions is often step 1, great scientific progress and often a huge leap forward. Then step 2 is often asking if there is some natural low-dimensional compression of the parameters that is approximately as good, and if so, does it correspond to any known quantities.
Though sometimes this isn’t possible or even desirable, such as with non-parametric models, and yet such models can still represent “interpretable” scientific progress in the sense of allowing us to predict how states of affairs will develop when we previously couldn’t and solving inference goals.
From a scientific perspective, the fact something can be modeled well using a neural network would provide interesting constraints on the system, which may eventually yield breakthroughs in understanding the original system. However, the neural network's parameterized equation by itself would never be accepted as good science, since its complexity would hardly contribute anything to our understanding.
People wouldn't waste time researching neural network visualization tools, coming up with adversarial examples and developing alternative (more explainable) machine learning models if that was the case.
But then in a practical setting, the model doesn’t correspond to something physical or to the inference goal of a practitioner. The p-value tells you something about the relative extremity of a certain statistic that, under certain assumptions, will have a particular distribution.
The practitioner wants to know the posterior probability of a particular model or hypothesis given the data, and the frequentist outcome literally can’t comment on it.
In this sense I think being able to state assumptions and connect them to outcomes with theorems is good, but not always necessary or even sufficient for applied work.
And there can creep in nasty subjective aspects of the problem that are uniquely defined by the specific inference goals at hand. Proofs about how a model would behave under assumptions are often totally useless in these cases. Practitioners don’t use linear regression for complex financial models because the assumptions hold or because of nice properties if the assumptions held. They use them because it’s simple and easy and sort of “just works” despite glaring flaws.