Bras cause breast cancer. Or so you'll discover in a new book, Dressed To Kill. One of the authors has an undergraduate degree in biology and a master's degree in anthropology. The other, his wife, is an optician.
Bras cause cancer? A story like this surfaces somewhere or other in the national press every week or two. Fields from power lines kill children. Breast implants cause arthritis. Herbicides cause infertility. Microwave ovens send Audis diving down elevator shafts.
Although laymen are sometimes swayed, scientists put very little stock in "discoveries" like these. Any number of different things can be fingered as suspect carcinogens - Alar, bras, caffeine, New Jersey and on down through the alphabet. But in many studies inquiring into the causes of cancer, few suspected links have proved out. And the few causes of cancer that have been pinned down - tobacco, diet and our bodies' own, natural hormones - are strong ones.
These facts engage a simple formula for inference elucidated by an 18th-century English mathematician, Thomas Bayes. In a nutshell: Inherently unlikely events remain unlikely even in the face of reports that they have occurred.
Consider a simple case. When your grandma sees a taxi, her eyesight is good enough to call the color right 80% of the time. If she reports seeing an orange taxi, how likely is it that she's got the color right? The answer is not 80%. It depends, and not on her eyes alone, or even mostly on her eyes, as you might suppose. The answer depends as much on records in the Bureau of Motor Vehicles. If 9 out of 10 taxis in the city are in fact yellow, and the rest orange, Granny's orange-taxi call will be wrong 9 times out of 13. If Granny has 50% accurate eyes and "sees an orange taxi," she will be wrong 9 times out of 10.
The bra-and-cancer stories always emerge from statistics much like these. There's a fallible observer - all observers are fallible. She claims to have identified one weak cause - bras, implants, power lines, or pesticides - of a common affliction, birth defects, colon cancer or arthritis. It has to be weak, because she (and we) know that arthritis is common even where implants aren't. Deconstruct this process and it always comes down to "seeing an orange taxi" in a city where most taxis are yellow.
Bayes' law tells us how to quantify the reliability of any new orange-taxi claim against background odds that heavily favor yellow. Correct observations of unlikely phenomena are themselves unlikely, even when coming from good eyes. Incorrect ones are horribly common. For a detailed discussion of this incontrovertible fact, take a look at Massimo Piattelli- Palmarini's delightful (but difficult) little book, Inevitable Illusions.
Bayes' law, though, is counterintuitive, which is why it is routinely overlooked by amateur scientists, like the ones who write popular books or script TV news shows. Correctly detecting one, comparatively rare, color (or cause) among a crowd of more common ones is very difficult indeed.
Think Bayes every time you see carcinogenic bras. A pair of perfectly sincere writers may see a bra behind every lump in a breast, just as Granny may report that all taxis are chartreuse. But Bayes establishes that the reliability of any such observation depends not just on the reliability of the observer ("good eyes"), but just as much on the underlying likelihood of the observation (yellow taxi or chartreuse).
Often this seems to beg the question: How can we get a handle on "background rates" if we don't know which observations to trust in the first place? Normally we don't have a Bureau of Motor Vehicles out there to hand us the second key bit of information that only 1 taxi in 1,000 is chartreuse; all we have is hundreds of grannies filing their individual reports. The best we can then do is ask: What do all other grannies - or in the case of bras and cancer, all other professional observers - have to say on the subject? If they report few, if any, chartreuse cabs, or carcinogenic bras, the odds against Granny's view of things, or your local optician's, rise fast.
So when the next pair of earnest unknowns publish a book advising you to burn bras, silicone, dioxin, PCBs or Alar, don't grab for a match too quickly. Bayes explains more than bras ever will. Most of the time, you're better off just burning the book.