Parade Magazine used to arrive in one of the local papers in Toronto when I was a grad student. At the time, I was busy trying to write new statistical software for problems for which there were as-yet no statistical solutions (minimum spanning trees, coevolution, etc). Famously, in 1990, Marilyn vos Savant answered a Parade reader’s inquiry about the Monty Hall problem. When I read it, at first it seemed to me as counter-intuitive as it proved to be for most of her readers. It turns out she was right if she is part of the collectivist Borg, and misunderstanding the position of a single human being. Read on.
The Monty Hall Problem
Monty invites you to pick one of 3 doors. He then opens one of the other two doors to show a door that does not have the prize. Monty invites you to stick with the door you first chose, or switch to the other unopened door. Marilyn vos Savant argued that your odds of winning are improved by switching. But the cause-and-effect logic-rat in your brain says “wtf?” In what universe can a choice made by someone else after I made my choice influence my choice between two unknowns?!
The power of learning to write code is the clarity with which you see your world. I too was confused by vos Savant’s assertion that you should switch your choice after Monty opens a dead-door. I decided to brute force the problem in the C programming language.
I didn’t even have to run my code (in Python, not C here) to prove the truth of her claim. In fact, it was in the midst of writing the core 6 or 7 lines of code that represent the decision tree that I saw it. Plain as day. The act of writing the code to solve the problem was itself my eureka moment. There’s no way to show you this without teaching you how to code, and having you actually go through the logic-centered process of writing the code to solve a problem.
This is why our daughters and sons need to learn how to code. It creates a clarity that cannot be accomplished even by reading how to code.
After writing less than 10 lines of code I was satisfied about the outcome, and I lost interest in finishing the program. After all, being the caontrarian I am, my motivation had been to prove it wrong, and I’d proved it right. Nonetheless, I labored on and finished the program in C, and then added all of the things to save a user from breaking it with unexpected input. I compiled it in a DOS environment and ran it and it worked. But by this point I knew it would.
In the Monty Hall Problem, switching away from the first choice leads to a 67% chance of winning. Remaining with the first choice results in a 33% chance of winning. I think the very best computational example of this was done as a version of Beer Hunter by James May.
But Popper was Right
There is an obvious disconnect between personal perception and statistical proof. The difference runs right through not only the Monty Hall Problem, but one’s choice of cancer therapy, and also what the effect will be if you get a vaccine shot against the pandemic coronavirus.
What vos Savant, and my little program widget did, was evaluate the outcome over hundreds or thousands of times playing Monty Hall’s little game. vos Savant’s assertion to “always switch your door choice” imputes from the law-of-large-numbers, down to the individual case. But, it is only Monty Hall who gets to play that game over and over. No individual contestant gets to do that. They will play once, and once only.
The solution to the Monty Hall Problem is not about what will happen to the contestant so much as what will happen to Monty Hall! If every contestant always switched, some would win everything, some would lose everything, and Monty Hall would still break-even. Should any percentage of contestants ever stay with their first choice, Monty Hall comes out ahead. This is the very definition of rigged casino game. There are saloons in the old West where this would have been illegal.
Karl Popper struggled with this, a lot! That is, that the frequency of some outcome in the case of the many, must necessarily describe some probability in the individual case. Why should we take it for granted that past outcomes dictate future outcomes? One would have to invoke what Popper called a “propensity interpretation”. That there is some law-like propensity that governs an individual event or choice that is properly described by the outcome of thousands of such choices.
Popper and I both knew
It was a confluence of my writing code to look at the Monty Hall Problem, and my delving deeper into the philosophical underpinnings of how we claim to know what we claim to know in evolutionary biology (without time machines) that led to my critique of Maximum Likelihood when I was a Michigan Society Fellow.
Fundamentally, like Marilyn vos Savant, the application of the method of Maximum Likelihood (and its probabilistic Bayesian cousin) requires that the frequencies we observe of events and things necessarily is imputable to single-events. Like speciation: whether chimps are our closest relatives or gorillas are.
What I tried to point out, perhaps inartfully, in Probabilism and Phylogenetic Inference, is that normally we are not concerned with the average outcome over hundreds or more species. We are not trying to be mostly correct. We are usually concerned with a very specific thing, like whether chimps are our closest relatives, or are gorillas.
My take on The Gambler’s Fallacy
Roberto Alomar is batting 0.300. He comes to bat three times in a game and fails to get a hit.
The naive gambler bets heavily on Alomar’s getting a hit on the fourth at-bat, because he is “due”. Our objective probabilist, like the likelihoodist, sees this differently and asserts that, because he is batting 0.300, he still has only a 30% chance of getting a hit, but this too fails to take into account the full scope of knowledge.
In the first place, because Alomar failed to get a hit in his last three times at bat, he is actually batting 0.297; the probabilities have changed, because they are historically contingent phenomena.
More to the point, Alomar either will or he will not get a hit and there is no probability that can be assigned to that one event: betting on one event alone is foolish.
My friend Jim Carpenter, a baseball fan, added this comically:
Actually, whether three at-bats drop Alomar’s average to 0.297 depends on where we are in the season. Early in the season, three 0-fer’s on a 0.300 average would drop the average a lot more, so his average would be well below 0.297, while late in the season the same sequence would drop the average less, so that he would still be above 0.297.
For illustrative numbers, consider that a 0.300 average the first week of the season would be based on something like three hits in 10 at-bats, so the collar drops him to 0.230, while the last day in the season, where a key starter like Alomar might have 600 at-bats, that means 180 hits gets 0.300, and the 0-for-three leaves him at 0.298 (with a chance to end at 0.300 — because it’s rounded up — if he gets a hit his last at-bat).
Which, contract incentives being what they are these days, he might really press to do, so if he hadn’t spat on any umpires recently you might bet on him to get that hit, because he’ll press a little, not too much, and late in the game he’s probably going to be seeing mostly fastballs anyway, even if it’s a blowout, because everybody will be looking to get out of there. All of which goes to prove the point about historical contingency.
And that is the point
Whether it is the Monty Hall problem, a casino, betting on baseball, or an HMO making decisions about who gets what cancer therapy and why, those decisions are being made by those who get to play this actuarial game over and over. Monty Hall comes back every day, the Casino can afford to lose many of the games it plays against you. And an insurance company, by definition is making money off of the idea that what is a rare-if-ever event for you is an every-day occurrence for them.
Vaccine efficacy
It is now widely known that the Pfizer and Moderna mRNA vaccines generate 95% protection from hospitalization after 2 shots, and as much as 80% protection after 1 shot.
Let’s dissect those statements. They are true of course, but they are statements about outcomes from clinical trials, and now population monitoring, over a very large group of people; not about one person’s expectation.
Let me begin by expressing what most people think it means and what it most assuredly does not mean. If you got only one shot of the Pfizer or Moderna vaccine, either you are fully protected or you are not fully protected. The 80% reported on one shot is about a population. It does NOT mean you as an individual are 80% protected. It does not mean that after one shot you will be infected 20% of the times you are exposed. I does not mean your immune system is at 80% capacity in fighting off COVID-19.
If you got only one shot of the Pfizer or the Moderna vaccine, either you are fully protected or you are not fully protected. It’s binary. And the odds are extremely likely that you are fully, and totally protected. Way better than a coin flip. The 80% number after one shot means after one shot 80% of the people who got it are fully and completely protected! Twenty percent of the people who got one shot and then got exposed, got sick. No one got 20% sick.
Cancer
In that same paper on Probabilism I noted that thought leaders in “other fields of science, including medicine, have already acknowledged that using statistical estimates … is unavoidably arbitrary, will often be contested and will have differential effects” upon our conclusions relating to singular cases.
Michael Hawley was one of my very best friends. Mike and I talked a lot about the various therapies at his disposal, and what the percentages looked like. I kept bringing him back to the notion that those percentages were about numbers of people, about wins for a hospital, or losses an insurance company. That they were not about him or his personal situation.
As we walked his two dogs around Cambridge, we went over the percentages he was reading, both of us knowing they had nothing to do with any percentage of years or days he would get to play with his infant son. They were about how many people like him would get to keep any days at all. We lost Michael last year to cancer. His cancer cheated him out of a lifetime with his boy… and cheated his boy out of a lifetime with a Renaissance Man. His therapies, that didn’t work, didn’t promise to give him some small percentage of those lost days, they promised him a small probability of having all of them. It was binary. Either they would work, or they would not. They didn’t. And there you go. And wherever you go, there you are.
One shot, two shot / this is not a flu shot
We need to vaccinate as many people as fast as possible. If we give more people the first shot and delay the second one, we are not creating a zombie-land of individuals who are incompletely vaccinated. We are creating a population in which 80% of the people you encounter are fully protected.
Let me say this very clearly: we live in communities of other people. It is our moral obligation to see 80% of more than 50% of the population protected over seeing 95% of 30% protected. It’s JUST math. The percentages are not about you, they are about all of us together.
Britain has already shown that this works better at driving down viral circulation. And Canada is following suit.