Sunday, March 3, 2013

Why the natural sciences rely on affirming the consequent (and that's OK!)

Last weekend, I discussed a paper by Graur et al., which (among many other criticisms and insults) accused the ENCODE consortium of basing certain interpretations on a fallacy in deductive logic called "affirming the consequent." I pointed out how bizarre that criticism was, because "affirming the consequent" is actually a necessary and justified part of reasoning in the natural sciences.

Many readers seemed to be surprised by and skeptical of this claim, and some probably thought it proof of my insanity. I must, first and foremost, once again beg such skeptics to read Jaynes' outstanding book. The first few chapters are actually available as a free pdf, but the whole book is really worthwhile. If you're an academic, you can probably find the book in your library system.

Understanding, however, that the urging of an apparent madman may not be adequate motivation, I thought I'd try to explain a bit more why this is, actually, the case.

tl;dr "affirming the consequent" is an error in deductive reasoning. It's spurious to raise this as an objection to findings in the natural sciences, because such findings are generally not based on deductive arguments. In fact, substantially all inferences in the natural sciences result from "affirming the consequent." This is because the natural sciences rely on confirmation and induction, epistemological processes under which "affirming the consequent" is permissible in a certain well-defined sense. This sense is nicely modeled by Bayesian probability theory, which can be viewed as a generalization of deductive logic to enable extrapolation and reasoning under uncertainty. When we attempt to restate Graur et al.'s objection using the proper vocabulary, we explicate the actual source of disagreement and avoid the need to level accusations of irrationality.

This lengthy essay will mainly discuss logic and inference in science. This is a really loose tangent from the whole ENCODE controversy, relevant only insofar as Graur et al. decided to make a big, unnecessary stink about it. So, you may well decide you have better things to do than read this, let alone Jaynes. But if it's not perfectly obvious to you why the "affirming the consequent" criticism is bogus, it's my sincere belief that you will benefit your scientific career by exploring this topic further.

Before diving in, let me also acknowledge some informed pushback on the other major problem I called out in my previous post about Graur et al., regarding definitions of "function" and the role of neutrally-evolving sequence elements in the genome. I'm going to write another blog post on that topic, but it may take me another few weekends! (Update: blogs by GENCODE and Nicolas Le Novère, and ironically Ford Doolittle's wonderful ENCODE critique, cover some similar ideas.)

Why the alarm?

You're on vacation, and you get an automated text message informing you that the burglar alarm at your home has been triggered. Has there been a burglary at your house? Should you call the cops?

Suppose that a burglary logically implies triggering of the alarm:
The problem with turning this around and inferring Burglary based on Alarm is that there are other possible causes of Alarm. Here in the San Francisco Bay Area, for example, small earthquakes happen once or twice a year, and the resulting vibration also causes Alarm.
Given Alarm, it's logically erroneous to infer Burglary, because Earthquake could also explain Alarm. Calling the cops when you see Alarm would make you guilty of the fallacy of "affirming the consequent" by the laws of deductive reasoning. Why are we paying for this stupid alarm again!?

This example is actually pretty tricky because I set it in the Bay Area, where small earthquakes happen more or less as often as burglaries of an individual home. Suppose instead that home is inner-city Detroit. Crime is high in that area, and earthquakes are quite rare. How, then, does our analysis change?

Actually, it doesn't change at all. Earthquakes are rare, but small ones do happen from time to time. Given Alarm, we cannot absolutely rule out Earthquake. So inferring Burglary from Alarm is still absolutely fallacious - affirming the consequent. That inference is simply illegal under the laws of deductive reasoning.

Of course, we have a gut feeling that deductive logic's answer is too rigid; there must be something more to the story. Indeed there is, and I'll get to that. But first, what does this have to do with science?

126 GeV? Pfffft.

In the natural sciences we attempt to learn general principles of the universe based on some observations we've collected. In particular, we seek to judge hypotheses and theories confirmed or falsified based on available data.

Occasionally we might find a datum that directly contradicts a hypothesis. That's rather useful, because then we can immediately declare that hypothesis falsified, and move on to others. There are difficulties with this in practice, though. For example, it's often trivial to surreptitiously fiddle with the original hypothesis just enough to resolve any contradiction. Also, when our data are obtained from elaborate instrumentation like particle colliders or microarrays, an apparent contradiction might well be due to an erroneous observation. These are actually crippling problems with "falsificationism" as an operational description of scientific inference, despite its enduring presence in the popular perception. (Not that falsifiability isn't an important, useful concept - it just ain't the whole story.)

A lot of other times, the data neither support nor flatly contradict a hypothesis. There's not much we can do then, although there are some journals for negative results, which can help save others' time and money.

What we most like to see, of course, is data that positively support our favored hypothesis - meaning that the hypothesis predicts or otherwise nicely explains the data. That's what gets papers published and grants funded! But, here's the rub: just as we can't logically infer Burglary from Alarm, no finite amount of data permits a deductive inference that a scientific hypothesis is generally true. No matter how many cases we've examined or what controls we've performed, there's always some possibility of an alternative explanation, or an instrumentation error, or a sampling error. So that $9B supercollider can never prove the existence of the Higgs boson. Similarly, that RNA-seq peak doesn't prove transcription, and transcription doesn't prove a "selected-effect" biological function; neither does conservation in a genome alignment, by the way.

Under the rules of deductive reasoning, we can never conclude a hypothesis is true unless we commit the fallacy of "affirming the consequent." Deductively speaking, progress in science implies "affirming the consequent." And clearly, science does in fact make progress. Ergo...

Don't just take my word for it

Now before you nitpick that argument - or call up MIT to demand the revocation of my PhD - take a look at the following quotes, all mined from credible sources on the interwebs:

Ha ha, I slipped a little Darwin in there on ya.

So, inference in the natural sciences seems to require that we commit the logical fallacy of "affirming the consequent." But as Darwin alluded to, and as I'll try to explain further below, this is nothing to worry about; in fact, it's just common sense!

One last thing: it certainly sounds pretty bad to be guilty of a logical fallacy, and in fact there are those who believe that "affirming the consequent" is a devastating criticism to level at scientists. Since I've just quoted Darwin, I'll bet you can guess who. That's right: creationists. Also, climate science deniers. Et cetera. This is a bread-and-butter argument for all of them.

Dear skeptical reader: is the horror beginning to set in?

About that burglar alarm...

Okay, if your house is in Detroit and your burglar alarm goes off, you'd better call the police. But this is logically fallacious - so how can we justify it?

Our common sense about this situation goes something like this: since Earthquakes happen so rarely, an Alarm is usually due to a Burglary.

One way to interpret how this works is to say that we're implicitly rounding the possibility of Earthquake down to nothing, so that we can pretend the alternative doesn't exist - thereby making the deductive inference of Burglary permissible. By rounding premises up to true, or down to false, we make the rules of deductive logic applicable.

But, doesn't it seem rather a shame to use the perfect laws of deductive logic on a mere approximation? Has it ever bothered you that such arguments never quite capture all the available information? What if it's not "often" or "rarely", but truly a toss-up, like in the Bay Area? Which way will we round then? Is it even possible to reason about that case in a rigorous way?

The answer is: yes, absolutely! Because there's a vastly richer interpretation of how common sense works in our example. In this interpretation, rather than rounding off the premises, we actually relax the laws of deductive logic to enable us to reason about degrees of belief in the two hypotheses. The system of reasoning we exercise by doing so is called Bayesian inference.

In our little example, applying Bayesian inference mainly entails dispensing with slippery words like "rarely" and "usually", and instead quantifying exactly how much more likely Burglary is than Earthquake as an explanation for Alarm in Detroit. In particular, Bayes' theorem provides a precise way to account for both the a priori plausibility of each hypothesis (Burglary and Earthquake) and also its power to explain the data (Alarm), in order to quantify the evidence favoring one hypothesis over the other. Having done so, we might conclude that we believe Burglary to be a better explanation than Earthquake for the Alarm by, perhaps, 100-fold. It's on this basis - that we believe there's only a very small, yet quantifiable risk of being wrong - that we can justify calling the cops.

There's tremendous depth I can't cover in this already-too-long essay, but here's the key insight: Bayesian inference is a generalization of deductive logic, able to account precisely for any degree of uncertainty. Both are sound mathematical theories of reasoning, which are related in an elegant way: when you plug zeroes and ones into the probability rules of Bayesian inference, some terms drop out and you get exactly the laws of deductive reasoning! In this strictly more powerful form of logic, some inferences that are absolutely illegal under the deductive rules become permissible in a certain well-defined sense. Namely, while Bayesian inference cannot prove any hypothesis that can't also be proven by deductive logic, unlike the latter it justifies increasing our degree of belief in a plausible hypothesis that explains the evidence at hand.

As our burglar alarm example illustrates, this approach to inference often captures "common sense" much more accurately than strict deductive logic - for rarely in everyday life do we stop to think about utter certainties. There are also other interpretations of "common sense" as it applies to our example, which are more precise than fudging the premises, but don't claim to be sound mathematical theories. Some terms you may have heard of include abductive reasoning, inference to the best explanation, Occam's razor, maximum a posteriori estimation, etc. These are arguably the most accurate explanations of "common sense" per se, since few of us actually go around calculating conditional probabilities in our heads. But Spock and Data are surely Bayesians!

Bayesian inference as a model of scientific reasoning

The Bayesian interpretation is a very powerful way to understand how scientific inference actually works - which is largely by common sense, supplemented with some best practices. Briefly, we evaluate our belief in a hypothesis based on our informed prior beliefs about its plausibility, on the one hand, and its ability to explain the available data, on the other. The data can never prove the hypothesis - rather, they may provide evidence in its favor. And while a hypothesis may thereby become preferred over any known alternatives, our degree of belief in it is always subject to future revision if we happen to come upon new information, or conceive of a new alternative. To give credit to falsificationism where due, it's exactly on such occasions that science makes the most progress.

Because Bayesian reasoning is closely aligned with common sense, most scientists write very well-reasoned papers without use of Bayesian statistical methods, without reference to Bayesian terminology, and even without Bayesian concepts in mind at all. Even if the author does happen to be an ardent Bayesian, style and space considerations usually do not permit formulating every sentence in a manuscript with the precise incantations of that system of reasoning. Occasionally, a word like "shows," "proves," or "demonstrates" will appear without a legalese qualifier like "beyond a reasonable doubt." Each such occurrence incurs a small bit of poetic license - and unambiguously commits the logical fallacy of "affirming the consequent." Usually, there's nothing terribly wrong with this.

To conclude this long exploration into logic and inference, I should acknowledge that Bayesian inference is certainly not all there is to scientific reasoning, but rather an excellent model of some of its main aspects. There's a lot of other creative stuff going on in the rational mind of a scientist, in the conceiving of hypotheses, the setup of experiments, the design of statistical models, and so on. Much of this has proven difficult to formalize so far, and some may never be. But insofar as it provides a sound, coherent solution to the key problems of confirmation and induction, it's not unfair to think of Bayesian inference as the essence of the scientific method.

Is that all Graur et al. really meant?

Let's (finally!) come back to the original motivation for this whole philosophical adventure. Graur et al. accused ENCODE of committing the logical fallacy of "affirming the consequent" by claiming certain genomic regions have biochemical functions based on data indicating specific biochemical signatures, such as transcription, transcription factor (TF) binding, chromatin modifications, etc. In the case of TF binding, they point out that it's logically erroneous to infer a function in regulating transcription, since TFs might bind without such effect. As we've discussed, this criticism is actually quite correct under the deductive rules called for by the terminology they chose, "affirming the consequent." The trouble is that if you accept it on those grounds, you also logically commit to invalidate substantially all inferences ever made in the natural sciences.

Have I just been way too anal this whole time? Maybe Graur et al. didn't intend for us to interpret their "affirming the consequent" criticism literally in terms of deductive logic, despite that term being completely specific to that theory. Perhaps all they really meant was that, given the evidence and their informed prior beliefs and definitions, there are alternatives at least as probable as the interpretations advanced by ENCODE. In this line of argument, ENCODE can't necessarily be accused of irrationality per se, but possibly of using a poor definition, or neglecting to account for alternative interpretations that others find highly plausible. Other commentators have raised such criticisms of the ENCODE paper, and frankly I think there is merit in them. In contrast, Graur et al. unmistakably chose to go further than this, and made the claim that even if you willingly grant ENCODE's definitions and premises, the conclusions are still wrong, because we committed the logical fallacy of "affirming the consequent."

Deductive reasoning is a rigorous mathematical theory. If you're going to attack others explicitly in the terms of that theory, your argument had better be defensible on those same grounds. In my opinion, anyone with a modern understanding of scientific inference just shouldn't level the criticism of "affirming the consequent" in those terms, because that criticism is itself definitely bogus. It seems valid, if you don't think about it too carefully - which is why it's popular among creationists and other science-deniers - but it logically commits you to absurd consequents. And by the way, Graur et al. don't just mention this in passing: it's fully explicated in a dedicated little section of the manuscript. Ouch - but at least it's not in the title.

Had they been expressed using the proper vocabulary as I sketched above, their objections to ENCODE's interpretations could have been stated formally, precisely, and coherently. Moreover, the actual source of disagreement - definitions, alternative hypotheses, and degrees of informed prior belief in the plausibility thereof - would have been obvious. Accusing ENCODE of a black-and-white logical error was a backfiring distraction from a more productive discussion about those stubbornly grey aspects.

Final words

I've gotta be perfectly honest - it took me about 10 years to clearly understand the principles underlying this essay. Probability was always a little bit mysterious to me as an undergraduate; I could do the calculations okay, but the meaning of it was elusive. I also took a few courses in philosophy of science, but there was limited rigor and quantification at that level. And once I got into research, the statistical methods prevalent in my field don't exactly help.

I'm sure many others "get it" much more quickly, but Jaynes' book was the key for me personally, and that's why I've been harping on it so much. The interpretation of probability as a generalized form of deductive logic is not an ancient or widely-appreciated insight, by the way; Jaynes was one of the first to recognize its importance, in the mid-20th century. Some debate continues to this day over a few of the strongest claims advanced in his writing, such as probability theory being the only coherent system of reasoning based on continuous measures of uncertainty. So while the book should not necessarily be taken as gospel in every detail, its explanations of Bayesian ways of thinking are unmatched in my experience.