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
ℙ[heads] = ℙ[tails] = 1/2.
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…
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.
This week’s leader in The Economist has me almost jumping up and down with joy, such is my enthusiastic approval. I am heartily sick of the childish name-calling that the political Left and Right indulge in, never more so than in the 2008 and present US Presidential campaigns. The divide (perhaps it is mutual incomprehension? or is it just rhetorical opportunism?) goes deeper than that, of course, but the current campaign and surrounding political climate throw it into stark relief.
“At the core, there is a failure of ideas. The right is still not convinced that inequality matters. The left’s default position is to raise income-tax rates for the wealthy and to increase spending still further—unwise when sluggish economies need to attract entrepreneurs and when governments, already far bigger than Roosevelt or Lloyd George could have imagined, are overburdened with promises of future largesse. A far more dramatic rethink is needed: call it True Progressivism.”
When I first became a regular reader of The Economist, some seven years ago, it was because I picked up a copy on a whim and was instantly bowled over by the pragmatic, decent, intelligent tone of the positions taken and how they were expressed. Since then, The Economist has certainly fallen short of that high mark for me several times. This week’s leader, though, is a hell of a return to form, the newspaper at its best.
And, of course, I’m not just admiring the rhetorical style, but heartily endorsing the position taken. As a species, we need political philosophies that rationally understand and take the best of both the social and selfish (competitive) sides of our human nature. We can do much better than to be one-sided extremists, and to portray opponents in the same light. So, if anyone feels like forming a “True Progressive Party”, then please count me in!
This article from the BBC makes for arresting reading. Bluntly put, the MPs flunked a very basic question in probability theory. If you’re a Bayesian (or other kind of subjectivist), then that finding literally makes those MPs irrational, stupid to the point of insanity. Less than a month after moves were made to end the “discriminatory” practice of mental illness being a bar to political office, we see that — provided that you identify sanity with rationality with logical reasoning — it is barely a bar as things stand at present! 😀
My tongue is only very slightly in my cheek when I ask this: Is there a way to use the results of the above numeracy survey as evidence for some kind of impeachment proceedings?
Several of my friends and acquaintances on the American side of the pond have remarked recently that they are fervently wishing for a break from the endless barrage of election coverage and the increasingly strident rhetoric emanating from each camp. In case closing eyes and clamping hands over ears isn’t enough, the following may serve as some solace: it’s not meant to be seen or heard in any reasonable sense anyway. As a letter-writer in the 25th August edition of The Economist remarks,
“This election will be very close and more voters than usual seem to have made up their minds already. The surest path to victory for either candidate is to motivate their core supporters.”
Put another way, it’s no longer about turning independent voters into Democratic-leaning or Republican-leaning ones (let alone flipping votes entirely!), but simply about keeping the emotions of the appropriately-inclined voters sufficiently inflamed that they actually go to the trouble of expressing their inclination at the polling booth in November. So, if the torrent of propaganda seems purely designed to irritate instead of persuade… that’s because it is! Knowing that, good luck to you all in tuning out the noise.
So much for mathematics: I seem to be on a bit of a politico-philosophical bent this week. Today’s article “Grand Racist Party?” in The Economist‘s “Democracy in America” series is rather interesting to me, for two main reasons plus one footnote.
First, citing Alex Tabarrok, John Sides and Reihan Salam, the article takes issue with the standard (at least on the USA’s political left) and simplistic assumption that “racist Americans are almost entirely in one political coalition [i.e. the Republican Party] and not the other”. The correlation is (a) weaker and (b) more complex than it is assumed to be, and it’s point (b) that I particularly like. I always like it when someone realises that assuming uniqueness is screwing up their understanding of a problem: it’s not racism, it’s racisms. Quoting Reihan Salam:
“[The] changing demographic composition of the U.S. population, and the changing cultural landscape, has given rise to other intercultural frictions, e.g., between non-Latino black Americans and Latinos, between non-Asians and Asians, etc. As we take into account these other forms of prejudice, one assumes that a very complex picture would emerge.”
In other words, when you try to map a whole multi-nodal network of interactions down onto a simple left-right political axis, you grossly over-simplify the problem and get it wrong. Straightforward white–v.–black prejudices might stand a chance of correlating well to a two-party split; the full complexity of American culture stands no such chance in my view.
Continue reading “Racisms, Liberties and Bulk Phenomena”
Whether or not sanity is statistical (as the interrogator O’Brien in George Orwell’s Nineteen Eighty-Four would have it), it is deeply political, even geo-political. What exactly constitutes sanity and whether or not certain critical individuals possess it are questions of great importance. Two examples of the loom large in my mind at the moment.
The first is the trial of Anders Behring Breivik, in which a verdict is expected within a week. The question is not one of the accused’s guilt, but of his sanity. Breivik seems happy to claim factual responsibility for the killings in Oslo and at Utøya Island; he claims moral culpability, but the court will award him that and sentence him accordingly if and only if it is satisfied that he is sane. The magnitude of Breivik’s crimes calls forth competing popular demands that he be regarded as insane (for what sane human could do something so vicious?) and sane (if nothing else, to deny him the “soft option” of psychiatric as opposed to penal incarceration). Breivik has articulated a rather lengthy and detailed philosophical and political position in justification of his acts. Is sanity the same as rationality? If so, then the question for those in the “he is insane” camp is this: is it his premises or his reasoning that are at fault? For me personally, the worrisome aspect is the “premises” side. But on to the second case…
Continue reading “The Politics of Sanity”