December 02, 2007

Quantum Causal Inference

Attention conservation notice: A strained conceit which involves two different mathematical sciences, and has at least one big problem even if you grant the outlandish major premise.

A basic part of modern ideas about causality and causal inference is what's sometimes called the "Neyman-Rubin counterfactual conception of causality" --- which is a mouthful, even if it does commemorate one of my heroes. The basic idea is simple: the causal effect of some difference in treatments on an outcome variable is the average difference between the value of the outcome between cases where one treatment was applied and those where the other one was applied to the same member of the population. The causal effect of a 419 e-mail on someone's net worth, for example, is the difference between their net worth if they answer the letter and if they do not. This is pretty obviously the kind of thing one wants to know about possible manipulations — though if you really want to get into the subtleties you should really read Glymour * and Pearl — but there is a problem: how can you both apply and not apply the treatment? How can you both answer and not answer the letter? (I am somewhat disappointed that there isn't an industry-funded P. T. Barnum Institute in Lagos, putting out press releases claiming that people who answered 419 letters would have lost even more money had they not done so.)

Trying to square this circle has lead to a lot of very ingenious statistical work, like propensity score methods and the consistent discovery algorithms of Spirtes, Glymour and Scheines. But now, via Wolfgang, I learn of something which makes me think that there is after all a physical solution:

Joseph Polchinski, "Weinberg's nonlinear quantum mechanics and the Einstein-Podolsky-Rosen paradox", Physical Review Letters 66 (1991): 397--400
Abstract: I show that Weinberg's nonlinear quantum mechanics leads either to communication via Einstein-Podolsky-Rosen correlations, or to communications between branches of the wave function.

Ruling out EPR correlations as unacceptable, let's concentrate on the bit about getting branches of the wave function to communicate with each other. Basically, Polchinski shows how to construct an apparatus where you make two measurements on the same particle in sequence, and the result of the second measurement depends on what you would have done had the first measurement been different. He turns this into a situation where "in effect, the apparatus reads the observer's mind", but what's more interesting to me is that this is just the kind of thing one would want to do for causal inference.

Here's how it goes. Step 1: Take a charged spin-1/2 particle like an electron. If we measure its spin along our favorite axis, say with a Stern-Gerlach device, we can get only one of two results, either +1/2 or -1/2, depending on the path the particle takes through the device. Do so, record the measurement, and rejoin the two paths. Step 2: If the spin was +1/2, do nothing. If the spin was -1/2, either (a) do nothing, or (b) rotate the spin with a magnetic field. Step 3: subject the particle to a field coupled to a suitably-constructed nonlinear observable; this is where the nonlinearity comes in. (I realize that sentence means nothing if you don't know some ordinary quantum mechanics; the point is, it's doing something that's not possible in ordinary QM.) Step 4: If you measured +1/2 in step 1, measure the spin again. Otherwise, do nothing. Upshot: If you take a measurement in step 4, you get +1/2 if you took action (a) in step 2, and -1/2 if you took action (b). (That's not obvious but does follow from Polchinski's calculations.) But to take either action, you'd have had to have measured a spin of -1/2 in step 1, which means you don't take a measurement at all in step 4. Thus, "the action and observation are in two different branches of the wave function".

Polchinski's set-up is a little too simple to estimate the causal effect of responding to a 419 scam, but that's just an implementation detail. Quantum mechanics seems very strongly to be a linear theory, but if that's wrong and it's really nonlinear (if only very slightly), then, pretty generically, doing causal inference should just be a matter of constructing the right sequence of measurements to act as an "Everett phone" between branches of the wave-function. Good-bye, covariate matching; hello, interferometry!

*: I can't resist quoting a paragraph.

I am tempted to think [that the demand] that a cause always be relative to a specific alternative is an improvement on the bare counterfactual account of causal relations. The reason is this: My Uncle Schlomo smoked two packs of cigarettes a day, and I am firmly convinced that smoking two packs of cigarettes a day caused him to get lung cancer. But it may not be true that in the closest possible world in which Uncle Schlomo did not smoke two packs a day, he did not contract cancer. Reflecting on Schlomo's addictive personality, and his general weakness of will, it may well be that the closest possible world in which Schlomo did not smoke two packs of cigarettes a day is a world in which he smoked three packs a day. I can reconcile this reflection with the counterfactual analysis of causality by supposing ... that "smoking two packs of cigarettes a day caused him to get lung cancer" is elliptical speech, and what is meant, but not said, is that smoking two packs of cigarettes a day, rather than not smoking at all, caused Schlomo to contract lung cancer.

Physics; Enigmas of Chance; Modest Proposals

Posted at December 02, 2007 15:20 | permanent link

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