Bayesian probability banned?

This post on Understanding Uncertainty bears the amusing, alarming and somewhat over-stated title “Court of Appeal bans Bayesian probability (and Sherlock Holmes)”. It’s not unusual for people to experience a little intellectual indigestion when first faced with the Bayesian probabilistic paradigm; it is particularly prevalent among people whose point of view is roughly speaking “frequentist”, even though they may have had no formal education in probability in their lives. Personally, I think that the judge’s criticisms, and Understanding Uncertainty‘s criticisms of those criticisms, are somewhat overblown.

However, I will advance one criticism of Bayesian probability as applied to practical situations. The basic axiom of the Bayesian paradigm is that one’s state of knowledge (or uncertainty) can be encapsulated in a unique, well-defined probability measure ℙ (the “prior”) on some sample space. Having done this, the only sensible way to update your probability measure (to produce a “posterior”) in light of new evidence is to condition it using Bayes’ rule — and I have no bone of contention with that theorem. My issue is with specifying a unique prior. If I believe that a coin is perfectly balanced, then I might be willing to commit to the prior ℙ for which

But can I really know that the coin is perfectly fair? Can I reasonably be expected to tell the difference between a perfectly fair coin and one for which

| ℙ[heads] − ℙ[tails] | ≤ 10−100?

(By Hoeffding’s inequality, to be satisfied with confidence level 1 − ε of the truth of this inequality would take of the order of 10100 (− log ε)1/2 / √2 (i.e. lots!) independent flips of the coin.) If not, then any prior distribution ℙ that satisfies this inequality should be a reasonable prior, and all the resulting posteriors are similarly reasonable conclusions. This kind of extended Bayesian point of view goes by the name of the robust Bayesian paradigm. It may seem that the difference between 10−100 and 0 is negligible… but it is not! The results of statistical tests can depend very sensitively on the assumptions made, especially when there is little data available to filter through those assumptions (and, scarily, sometimes even in the limit of infinite data!).

So, yes, I agree that (classical) Bayesian statistics shouldn’t be let near life-or-death cases in a courtroom. But robust Bayesian statistics? I could support that…

Null vectors and spinors

My recent reading on the topic of spin and angular momentum in quantum mechanics led me to the concept of a spinor. It turns out that spinors are fearsomely nasty objects to wrap one’s head around in full generality, requiring Clifford algebras and other ingredients, although the three-dimensional case is quite accessible and is described below. The (slightly unhelpful) heuristic is that spinors behave like vectors except that they change sign under rotation through an angle of 2π — a somewhat confusing property that will be made clearer later.

First, some general notions: let V be a vector space over a field K, equipped with a bilinear map b: V × VK and hence a quadratic form q: VK given by q(v) ≔ b(v, v) for all v ∈ V. A vector vV is called a null vector or isotropic if q(v) = 0. Recall that the standard Euclidean bilinear form on ℝn is

$\displaystyle b \bigl( (x_{1}, \dots, x_{n}) , (x'_{1}, \dots, x'_{n}) \bigr) \equiv x \cdot x' := \sum_{j = 1}^{n} x_{j} x'_{j} .$

This bilinear form has no non-trivial null vectors (i.e. the only null vector is the zero vector), but two close relatives of the Euclidean bilinear form do have interesting null vectors.

[Somehow, WordPress deleted large chunks of this post. Apologies! If the text below differs from what you saw earlier, then blame the post-deletion restoration effort.]

Sarah Mayer

Sarah Mayer was the first non-japanese woman in the world to be awarded black belt rank in Kodokan Judo, an achievement that made the headlines of The Japanese Times on 1 March 1935. During her two-year stay in Japan, she wrote several letters home to Gunji Koizumi, Founder of the Budokwai dojo in London where she had begun her training. These reproductions of the letters are well worth reading as memoirs of martial arts practice in Japan in the inter-war period, especially for someone who would have stood out from the vast majority of practitioners in two ways, as a woman and as a gaijin (non-Japanese). They are written with a wonderfully understated dry English wit, perhaps exemplified by the passage

[A] strapping young man of 5th Dan had been called in to practice with me. For awhile we pranced around and he let me throw him about a bit and dropped me fairly gently on the mat and then the Professor said something to him and he threw me all over the place, and not content with throwing me, he gave me that extra push when I was on my way down that makes the floor come up quicker than usual. […] I was beginning to think that it was too much of a good thing and to wonder how best I might escape from his clutches without letting down the British Empire by asking him to be a bit less rough with me, when it occurred to me that although I was being thrown with some violence, I had not yet hurt myself, so I decided that it would be better to wait until I died before I complained.

The true nature of the beast

A bit of dialogue that rings in my head a lot, from the movie Crimson Tide:

Ramsey: At the Naval War College it was metallurgy and nuclear reactors, not Nineteenth Century philosophy. “War is a continuation of politics by other means.” Von Clausewitz.
Hunter: I think, sir, that what he was actually trying to say was a little more —
Ramsey: Complicated? [Men laughing]
Hunter: Yes, the purpose of war is to serve a political end, but the true nature of war is to serve itself.
Ramsey: [Laughing] I’m very impressed. In other words, the sailor most likely to win the war is the one most willing to part company with the politicians and ignore everything except the destruction of the enemy. You’d agree with that?
Hunter: I’d agree that, um, that’s what Clausewitz was trying to say.
Ramsey: But you wouldn’t agree with it?
Hunter: No, sir, I do not. No, I just think that in the nuclear world the true enemy can’t be destroyed.
Ramsey: [Chuckling, tapping glass] Attention on deck. Von Clausewitz will now tell us exactly who the real enemy is. [Laughing] Von? [Men laughing]
Hunter: In my humble opinion… in the nuclear world… the true enemy is war itself.

These are some notes, mostly for my own benefit, on annihilation, creation, and ladder operators in quantum mechanics, with a few remarks towards the end on angular momentum, spin and Clebsch–Gordan coefficients.

First, the abstract definition: if T, LV → V are linear operators on a vector space V over a field K, then L is said to be a ladder operator for T if there is a scalar cK such that the commutator of T and L satisfies

$\displaystyle [T, L] := TL - LT = cL.$

The operator L is called a raising operator for T if c is real and positive, and a lowering operator for T if c is real and negative.

The motivation behind this definition is that if (λv) ∈ K × V is an eigenpair for T (i.e. Tv = λv), then a quick calculation reveals that (λ + cLv) is an eigenpair for T:

$\displaystyle T(Lv) = (TL)v = (LT - [T,L])v = LTv + cLv = (\lambda + c) (L v).$

Ladder operators come up in quantum mechanics because many of the elementary operations on quantum systems act as ladder operators and increase or decrease the eigenvalues of other operators. Those eigenvalues often encode important information about the system, and the increments and decrements provided by the ladder operators often come in discrete, rather than continuous, values. Annihilation and creation operators are a prime example of this phenomenon.

Knee strengthening exercises

Just a short post today to recommend this article on knee strengthening exercises by Martin Koban. Obviously, exercises to strengthen and stabilize the knee are of great interest to me at the moment as I recover from my knee surgery. I’d say that I was probably already familiar with 50% of these exercises, and look I forward to working up to doing them all. Hopefully, consistent practice of these exercises will stave off future knee issues.

Interpolation and fractional differentiability revisited

In this earlier post on interpolation spaces, part of the motivation for studying interpolation spaces was the search for a reasonable space of functions with a non-integer order of differentiability 0 < α < 1. In the case of strong (classical) derivatives, a suitable such space was the vector space Cα(K) of α-Hölder functions on a compact set K ⊆ ℝn with interior, i.e. the set of functions uK → ℝ for which the norm

$\displaystyle \| u \|_{C^{\alpha}(K)} := \| u \|_{\infty} + \sup_{\substack{ x, y \in K \\ x \neq y }} \frac{| u(x) - u(y) |}{| x - y |^{\alpha}}$

is finite. In the case of weak derivatives, a suitable such space was the vector space Wα,p(K) of functions u for which the norm

$\displaystyle \| u \|_{W^{\alpha, p}(K)} := \left( \| u \|_{L^{p}(K)}^{p} + \iint_{K} \frac{| u(x) - u(y) |^{p}}{| x - y |^{\alpha p + n}} \, \mathrm{d}x \mathrm{d}y \right)^{1/p}$

is finite. These spaces are all Banach spaces, and interpolate in the sense of real K-interpolation between the spaces C0(K) of continuous functions and C1(K) of continuously differentiable functions (respectively the Lebesgue space Lp(K) and the Sobolev space W1,p(K). This post grew out of my noticing one simple omission in the previous post, now corrected: for p = 2, the spaces Wα,2 are Hilbert spaces under the inner product

$\displaystyle (u, v)_{W^{\alpha, 2}(K)} := \int_{K} u(x) v(x) \, \mathrm{d} x + \iint_{K} \frac{( u(x) - u(y)) (v(x) - v(y))}{| x - y |^{2 \alpha + n}} \, \mathrm{d}x \mathrm{d}y .$

On realizing this omission, I started to think more deeply about other notions of fractional differentiability. In particular, I wondered how the above Wα,p spaces are related to other fractional-order Sobolev spaces defined using Fourier transforms. So, the rest of this post is devoted to surveying the two main methods of constructing fractional-order Sobolev spaces and the relationships between them.

For neatly self-contained proofs of the assertions in this post, I recommend this set of notes by Eleonora Di Nezza, Giampiero Palatucci and Enrico Valdinoci.