Xiao-Li Meng writes:
Some events in life repeat, although not necessarily in an i.i.d.* fashion. For some unlucky or lucky ones (see my last XL-Files) this has been a season of repeated storms and flu. A trip to London had to be canceled because the flu bug, with her fever(ish) sister, visited me in mid-December, providing me with a much-needed excuse to stay in bed. I even had time to contemplate how faithful is an English translation of the poem that has touched billions of feverish (Chinese) hearts, By Chance (偶然):
“I am a cloud in the sky / A chance shadow on the wave of your heart / Don’t be surprised / Or too elated / In an instant I shall vanish without trace…”
But the storms took away the cloud, and the feverish sisters certainly did not vanish in an instant.
As a matter of the fact, they put me in bed again, right after the New Year and just before another trip to London (and then China). Thank God that the i.i.d. assumption did not seem to apply, because my body did learn from its last encounter with the fiendish sisters. Consequently, I had time only to finish “My Brief History,” and with no time to get to “The Briefer History of Time,” I developed a deeper appreciation for Stephen Hawking’s desire, and ability, to construct brief/briefer time.
Of course n=2 does not qualify for being frequent(ist), especially when the effective sample size is less than two, thanks to the shadow or “latent trace.” However, the flu bug, this time without her feverish sister, waited for me at Logan airport upon my return from China. Now this got a bit too tiring, especially because I had another guest with me—Mr. Jetlag. Indeed, I was so fatigued that I started to wonder if I had some other uninvited guests living with me. I therefore visited my doctor, who ordered a number of blood tests, which all came back negative. Great! Forget about all the test sensitivities and specificities that I emphasized so much when teaching “Vital Statistics for Life and Medical Sciences”; surely, I had no interest to second guess the results that I had hoped for.
But sometimes hope alone does not do the trick — I was still feeling very fatigued. My doctor examined my medical history, and found that once I was diagnosed with a mild sleep disorder (apparently, my body is a bit too active when it should enjoy a downtime). Upon learning that I had not been properly treated (as I did not think that was necessary), he smiled: “Well, that explains it. Guess what’s the #1 symptom of having a sleep disorder? Fatigue! I am therefore almost certain that you will feel better if you get it treated.”
I instantly felt more energy, for I found the reason and a cure. But the moment I walked into the snow, the statistical flakes cooled me down: “Wait a minute… he just played a fiducial trick on me!” It’s a classical “fiducial switching”: that (1) fatigue, F, is the #1 symptom, S, given sleep disorder, SD, as the cause, C, implies that (2) SD is the #1 cause for F.
This assertion is not guaranteed even if we allow ourselves to commit a prosecutor’s fallacy by interpreting P(S|C) as P(C|S), because for (1) and (2) to hold simultaneously would require (F, SD) to be the most frequent combination among all pairs of (S, C) that contain F or SD. Furthermore, the prior P(C=SD) is not necessarily 1 even if I have SD, unless we can rule out competing risk.
Indeed, I was previously told by a doctor at University of Chicago that I had CFS (chronic fatigue syndrome), for which there is no blood test. Surely P(S=F|C=CFS)=1≥P(S=F|C=SD), and to make the matter worse, a common symptom of CFS is SD. So now should I still be treated for SD? Incidentally, another doctor at Chicago thought CFS should really stand for “Chicago faculty syndrome,” because he attributed all my fatigue to my anxiety about getting tenure. (My fatigue did go away shortly after I got tenure. Coincidence?)
My doctor may well have reasoned correctly based on the information he had, and he chose to convey it to me in a simplistic way, unaware of my occupation. But the very fact that my statistically well-trained (or apparently not!) mind was so eager to accept the “fiducial cure” highlights an issue that has been largely unformulated in statistical/probabilistic literature: the impact of an outcome’s desirability (or lack of it) on our ability to rationally assess or interpret its probability or risk. Fortunately, our philosophy friends take such problems to heart. Check out books such as Lara Buchak’s Risk and Rationality (2013) during your next feverish encounter, though of course I don’t wish you to have to finish an entire book in bed!
*For any accidental visitors to XL-Files (looking for X-Files?), “i.i.d.” stands for independently and identically distributed, which in plain English means events following an identical pattern yet providing no information about each other.