[*Update*: I have fixed some broken links, and modified the discussion of QBism and the recent paper by Chris Fuchs— thanks to Chris himself for taking the time to read and find my mistakes!]

For some reason, I’ve come across an idea called “Knightian Uncertainty” quite a bit lately. Frank Knight was an economist of the free-market conservative “Chicago School”, who considered various concepts related to probability in a book called Risk, Uncertainty, and Profit. He distinguished between “risk”, which he defined as applying to events to which we can assign a numerical probability, and “uncertainty”, to those events about which we know so little that we don’t even have a probability to assign, or indeed those events whose possibility we didn’t even contemplate until they occurred. In Rumsfeldian language, “risk” applies to “known unknowns”, and “uncertainty” to “unknown unknowns”. Or, as Nicholas Taleb put it, “risk” is about “white swans”, while “uncertainty” is about those unexpected “black swans”.

(As a linguistic aside, to me, “uncertainty” seems a milder term than “risk”, and so the naming of the concepts is backwards.)

Actually, there are a couple of slightly different concepts at play here. The black swans or unknown-unknowns are events that one wouldn’t have known enough about to even include in the probabilities being assigned. This is much more severe than those events that one knows about, but for which one doesn’t have a good probability to assign.

And the important word here is “assign”. Probabilities are not something out there in nature, but in our heads. So what should a Bayesian make of these sorts of uncertainty? By definition, they can’t be used in Bayes’ theorem, which requires specifying a probability distribution. Bayesian theory is all about making models of the world: we posit a mechanism and possible outcomes, and assign probabilities to the parts of the model that we don’t know about.

So I think the two different types of Knightian uncertainty have quite a different role here. In the case where we know that some event is possible, but we don’t really know what probabilities to assign to it, we at least have a starting point. If our model is broad enough, then enough data will allow us to measure the parameters that describe it. For example, in recent years people have started to realise that the frequencies of rare, catastrophic events (financial crashes, earthquakes, etc.) are very often well described by so-called power-law distributions. These assign much greater probabilities to such events than more typical Gaussian (bell-shaped curve) distributions; the shorthand for this is that power-law distributions have much heavier tails than Gaussians. As long as our model includes the possibility of these heavy tails, we should be able to make predictions based on data, although very often those predictions won’t be very precise.

But the “black swan” problem is much worse: these are possibilities that we don’t even know enough about to consider in our model. Almost by definition, one can’t say anything at all about this sort of uncertainty. But what one must do is be open-minded enough to adjust our models in the face of new data: we can’t predict the black swan, but we should expand the model after we’ve seen the first one (and perhaps revise our model for other waterfowl to allow more varieties!). In more traditional scientific settings, involving measurements with errors, this is even more difficult: a seemingly anomalous result, not allowed in the model, may be due to some mistake in the experimental setup or in our characterisation of the probabilities of those inevitable errors (perhaps they should be described by heavy-tailed power laws, rather than Gaussian distributions as above).

I first came across the concept as an oblique reference in a recent paper by Chris Fuchs, writing about his idea of QBism (or see here for a more philosophically-oriented discussion), an interpretation of quantum mechanics that takes seriously the Bayesian principle that all probabilities are about our knowledge of the world, rather than the world itself (which is a discussion for another day). He tentatively opined that the probabilities in quantum mechanics are themselves “Knightian”, referring not to a reading of Knight himself but to some recent, and to me frankly bizarre, ideas from Scott Aaronson, discussed in his paper, The Ghost in the Quantum Turing Machine, and an accompanying blog post, trying to base something like “free will” (a term he explicitly does not apply to this idea, however) on the possibility of our brains having so-called “freebits”, quantum states whose probabilities are essentially uncorrelated with anything else in the Universe. This arises from what is to me a mistaken desire to equate “freedom” with complete unpredictability. My take on free will is instead aligned with that of Daniel Dennett, at least the version from his Consciousness Explained from the early 1990s, as I haven’t yet had the chance to read his recent From Bacteria to Bach and Back: a perfectly deterministic (or quantum mechanically random, even allowing for the statistical correlations that Aaronson wants to be rid of) version of free will is completely sensible, and indeed may be the only kind of free will worth having.

Fuchs himself tentatively uses Aaronson’s “Knightian Freedom” to refer to his own idea

that nature does what it wants, without a mechanism underneath, and without any “hidden hand” of the likes of Richard von Mises’s

Kollectiveor Karl Popper’spropensitiesor David Lewis’sobjective chances, or indeed any conception that would diminish the autonomy of nature’s events,

which I think is an attempt (and which I admit I don’t completely understand) to remove the probabilities of quantum mechanics entirely from any mechanistic account of physical systems, despite the incredible success of those probabilities in predicting the outcomes of experiments and other observations of quantum mechanical systems. I’m not quite sure this is what either Knight nor Aaronson had in mind with their use of “uncertainty” (or “freedom”), since at least in quantum mechanics, we do know what probabilities to assign, given certain other personal (as Fuchs would have it) information about the system. My Bayesian predilections make me sympathetic with this idea, but then I struggle to understand what, exactly, quantum mechanics has taught us about the world: why do the predictions of quantum mechanics work?

When I’m not thinking about physics, for the last year or so my mind has been occupied with politics, so I was amused to see Knightian Uncertainty crop up in a New Yorker article about Trump’s effect on the stock market:

Still, in economics there’s a famous distinction, developed by the great Chicago economist Frank Knight, between risk and uncertainty. Risk is when you don’t know exactly what will happen but nonetheless have a sense of the possibilities and their relative likelihood. Uncertainty is when you’re so unsure about the future that you have no way of calculating how likely various outcomes are. Business is betting that Trump is risky but not uncertain–he may shake things up, but he isn’t going to blow them up. What they’re not taking seriously is the possibility that Trump may be willing to do things–like start a trade war with China or a real war with Iran–whose outcomes would be truly uncertain.

It’s a pretty low bar, but we can only hope.