Dr. X: Hi, Joe! Here are your test results — notice that the first two are not actually tests you took, but you can use them the same way — and here’s Jack, a nurse who’s learning to explain Bayesian reasoning because that really wasn’t covered well in nursing school. He has paperwork or rather computer work to do, but he’ll answer questions, and I’ll see you in a little while — Bye-for-now! [Turns away, turns back.] I promise that if I think things are getting really urgent I’ll tell you right away. [Goes.]
Joe: but… but…
Jack: It’s okay, I’ll work in here. [sits.] Most of your questions you’ll probably want to address to the duck, though, so just say “Jack!” before you ask anything of me because otherwise I probably won’t be listening.
Joe: the duck?
Jack: Yes, this perfectly ordinary plastic bath-toy duck. [QUACK-QUACK!] It’s an old trick — to get your thoughts in order and answer most of your own questions, just ask the duck, then think — did that question even make sense? Often you’ll realize that to ask the right question you need to figure out something else first, and once you finally do ask the right question it answers itself. But you usually have to ask out loud. So a helpful rule is sometimes to have somebody like me that might be able to answer an actual question once you’ve figured out the question that you really need to ask, but every such question has to be asked twice: you ask the duck, and only if you’re sure you can’t improve that question on your own do you repeat it with your helper’s name in front.
Joe: I don’t understand what’s going on here. Do I have Ickitis or not? [waits]. Oh, right. Jack, do I have Ickitis or not?
Jack: That’s a very good question, and the answer to that question is the same as it is for a whole lot of questions: we don’t know for 100%-certain-sure, but we can say something about
[1 finger] how likely it is that you have it, and about
[2 fingers] what will likely happen if you have it and don’t treat it, and about
[3 fingers] how likely it is that you’ll have bad effects from treating it even if you don’t have it.
Then you have to choose what comes next — or you can go back to asking Dr. X what to do. He said you came in after getting a test for yourself, which a lot of people do, and that you were willing and able to think about it, which a few people are, and so maybe you’re the kind of patient that can learn to do this kind of thinking. If so, he’d really like to encourage that, but if not, we’ll go back to traditional practice. Okay?
Joe: Umm, yeah, I guess. I mean, yeah! I’ll try. So what’s this handout here? No, I mean, never mind. Duck, what’s this handout? I guess you don’t know, so I’d better read it…written specially for me? Umm… Jack, is this written specially for me?
Jack: Well, I filled in some blanks for you from Dr. X’s note, with the “ickitis” name and the odds ratios for the ickitis test you got for yourself, and for the next test which you can read aloud to the duck — I think it might make sense that way.
Joe: Oh. Okay. [clears throat]. Duck, “You started with yes-to-no odds of one to 999 in favor of having ickitis, and took a test that said you had it. The test had right-to-wrong odds of 99 to 1, and that gave you new yes-to-no odds of (1/999) * (99/1) or 99 to 999 that you had it, which was 11 to 111 or 11 chances out of 122 which is about 9%. And you know how to interpret that by visualizing four bunches of people: the infected who test positive or negative, and the uninfected who test positive or negative. There were 1000 infected with 990 of them testing positive, and 999,000 uninfected with 9,990 testing positive anyway.” Okay, duckie, I remember that.
Joe: Duck, “Test#2 is not a medical test. You’re asymptomatic. Most infected people are not asymptomatic, and most infected people are. With ickitis, 60% of infected people have symptoms, 40% don’t, and we call that a sensitivity of 60%. 5% of uninfected people have symptoms anyway, presumably from being infected by something else that could be confused with ickitis, and 95% don’t. We call that a specificity of 95%.”
Joe: Duck, that actually makes sense. But I have no idea what to do with it except to go on reading. “You can use those proportions to update your probability. Imagine four groups of people arranged in rows and columns: 122 columns of 100 rows. The first 11 columns are of infected people, the remaining 111 are the uninfected, because a random person in that array starts out to be just as likely as you are to be infected. Among the infected, the first 60 rows do show symptoms and the rest don’t: that’s the sensitivity. Among the uninfected, the first 95 rows don’t show symptoms and the rest do: that’s the specificity. If you don’t show symptoms, what is the probability that you’re actually infected?”
Joe: Duck, I think maybe I can do this. Let me think.
Daily Bayes 