The 'Statistician' Loved by Silver
We could let N go to the limit immediately (before taking observations). Now no real thing is infinite. There never has been, nor will there ever be, an infinite number of anything, hence there cannot be an infinite number of observations of anything. In other words, there is just is no real need for notions of infinity. Not for any problem met by man. Which means we'll go to infinity anyway, but only to please the mathematicians who must turn probability into math.
— William Briggs, from Subjective Versus Objective Bayes (Versus Frequentism): Part Final: Parameters!
I'll get to that paper in a bit. But first, let's parse what Briggs's claim. '[T]here is just is no real need for notions of infinity.' Kronecker would be proud. But our notions about these things have advanced a bit since the 1800s. And even if they hadn't, sometimes it can be useful to use continuous approximations, or to take limits. So even if you think the universe is discrete on all levels (Planck lengths and Planck times, and all that physics), we can still find uses for the infinite.
So, Briggs has a disdain for mathematics. Which I would have once thought was an odd opinion for a statistician to have. But I've since learned that some want to turn statistics into an empirical science. Which seems a bit daft to me. Like a dog chasing its tail.
Back to his paper.
Since none but the devout want math, skip it or minimize it. We will not lack for students: even without a saturation of mathematics, those who hear the call will still enter the fold. Since civilians want to understand uncertainty, teach it. This is our great forgotten purpose.
I've been known to claim that we should only teach statistics to those who have earned a black belt in mathematics. Statistics is hard, subtle, and counterintuitive. But it's rarely taught that way. Students taking a STAT100-style course are, of course, already intimidated by basic arithmetic. But I don't think we should water down the material. Instead, we should improve basic math education so that students might rise to the occasion.
At the same time, Briggs notes that we need to teach these people something if they have any hope of getting along in quantitative fields (or even just in society). I don't have a good answer for this. I feel like more harm than good is often done with introductory statistics classes. But I don't think students would leave a course centered around Bayesian statistics with all that much more subtle of a view on the subject.
Recovery of these courses is important because there are lot of folks out there who, because they once had a graduate survey course in regression, and have personally produced a p-value or two, feel they are versed sufficiently in probability to pass on their skills to fresh students. But their rendering of the subtleties of the subject is often not unlike the playing of a worn cassette tape, which itself was copied from a bootleg. This might even be acceptable, except that some who hear these concerts go on themselves to teach statistics. This is an odd situation. It is as if a cadre of mathematicians who had once read a greeting card and then wrote a love poem felt that they were qualified to teach English Literature and then began doing so—and issued credits for it, too—all with the hope that their students will go on to write novels.
This I couldn't agree with more.
We have a subtle twist on this problem in our own department. The intermediate level statistics classes, such as STAT400, are mostly taught by probabilists, not statisticians. Take the professor for this upcoming semester, for example.
There's a huge difference between a probabilist and a statistician. Both use probability, sure. But a probabilist is more likely to care about the finer points of measure theory than how to munge data into a useable format. We should have probabilists teaching probability courses, and statisticians teaching statistics courses. This seems like common sense: we rarely have physicists teaching mathematics courses or vice versa. But our statistics department is nearly non-existent, so generations of students are passing through statistics classes run by professors who, honestly, probably don't really care for statistics.
The problem is that we only have seven statisticians on staff. Only five of which really count as statisticians (I would count Freidlin and Lee as probabilists).
Statistics (and mathematics) education is certainly not a solved problem. I know their are legions of researchers out there exploring this very question. (The fact that this literature is so rarely read by people like me who claim to care about these things is another problem.)
But replacing frequentist methods with Bayesian methods certainly isn't a solution.