Personal essays
Thoughts on Bayes’ Theorem
Gather together
Thanksgiving isn’t just about the turkey, stuffing, and other sides.
“Of course not Daniel,” you say, “it’s about the Chex Mix.”
True. But it’s also about the people. My favorite memories of Thanksgiving are of a house filled with people. The meal was kind of the stone in the stone soup story. It’s the excuse for everyone to gather.
Each year magazines come out with “Ten Tantalizing Turkey Techniques” of “The hot new side dishes for your holiday table.”
Nonsense. The traditional dinner serves as a conduit to the past. That same turkey and stuffing (or dressing in my house) is like a wormhole that allows you to remember back to Thanksgivings past and prepares the next generation to host Thanksgivings future.
For me, it’s about being home - preferably my home - with food being cooked - preferably by me - for a house filled with people.
And so after spending six of the last eight weeks on the road, I’m home for Thanksgiving.
I don’t love the travel part so much as I love the being there.
There is so much about these two trips that has been great - I’ve spent time in Spain, France, Italy, The Netherlands, England, and Scotland.
Even better, I’ve spent time with friends from all over the world.
As much as I love being home and seeing my friends and family here, after decades of traveling and conferences, I have so many people I care about that live far far away.
Who’s in
One of the cool things when traveling is arranging to meat people for meals.
There are unexpected meetings.
I got an email from a group in London announcing a meetup featuring a friend of mine who I didn’t know was going to be in town. It was like a gift. We were both free for dinner the night before the meetup and so I had a wonderful an unexpected meal with someone I wasn’t expecting.
More often I contact or am contacted by people who live in the towns I’m visiting and we arrange to meet. My weekend in Edinburgh was like that. When someone had to cancel, I was able to extend my time with another friend which meant our discussion went deeper.
They had chosen the topic based on last week’s essay on fairness and argued that fairness doesn’t scale.
We talked about a ton of other things but I love that I have friends that want to chat about things like that (and calculus, and apps we’re thinking about, and dating, …) but the topic that stuck with me was: fairness doesn’t scale.
I don’t know.
So I thought about this scheduling of encounters and a situation that is like scaling that up.
Not literally - not scheduling who people meet with in a society - but imagine this:
Imagine our country has mostly good people in it but there are some bad people as well. For concreteness, let’s say that 99% of the people are good people.
“Oh Daniel, 99% of the people couldn’t be good people,” you say.
Well, let’s say that 99% of the people are good enough. (In fact, I’m thinking of the percent of people who vote legitimately compared with the percent who commit voter fraud so actually the percent is much closer to 100%.)
And let’s say we can devise a test that will help us keep the good people and send away the bad people. Suppose that this test is 90% accurate.
So if we give the test to 100 good people, on average the test correctly reports that we should keep 90 of them and incorrectly reports we should send away 10 of them even though they are good.
We might express this as “the probability we keep a person given that they are good is 90% or 0.9.”
| We would write this as P(K | G) = 0.9. |
Don’t freak out - there’s not going to be much math here.
And if we give the test to 100 bad people, on average the test correctly reports we should send away 90 of them and incorrectly reports we should keep 10 of the bad ones.
| P( S | B ) = 0.9. |
We love this test. It’s almost always right. We could develop a better test - but for now 90% feels good enough.
The problem
Bayes’ theorem allows us to turn the conditional around. It says, look at all of the people you’ve decided to send away. What percent are good and what percent are bad?
Suppose our country has 1000 people.
Then 990 of them are good and 10 of them are bad.
We give our test to the 990 good people and it says keep 90% of them so we keep 891 of them and send away 99 of them.
We give our test to the 10 bad people and it says send away 90% of them. So we send away 9 of them and keep 1 of them.
So all in all we keep 892 people and send away 108.
We’re feeling pretty good. Only one of those 892 people we keep aren’t good.
Hang on. Take a look at the people being sent away. Of the 108 being sent away, 99 of them are good and 9 of them are bad.
If you’re sent away, there’s a much better chance of you being a good person than being a bad person.
To put this in numbers, the probability that you’re a good person given that you’re one of the people being sent away is
| P (G | S) = 99/108 = 92%. |
It’s safe - the numbers are gone
It’s so tempting to devise tests for figuring out when someone is doing something wrong. If the percentage of the population doing something wrong is small enough, our test is more likely to reject people who shouldn’t be rejected than people that should be.
So we are taught to fear that these problems are wide spread.
Voter fraud is extremely rare. But if our leaders can convince us that it’s more common than we think then they can purge the voter roles. And the test they use to purge the roles is not 90% accurate and it is usually skewed against those with low income or people of color. And so for every ten thousand people whose names are struck off the roles, one may indeed be someone who should be but we’ve burdened the other 999.
“Hey, you said no more numbers.”
You’re right.
I just worry.
Between this Thanksgiving and next Thanksgiving so much could change in my country.
One of the presumptive candidates for president is already making plans for tests to decide who stays and who goes and he doesn’t seem particularly concerned if a lot of good people are scooped up and put in the send away bucket.
Can fairness scale?
I don’t know.
But I do know these “stay or go” tests can and do scale and we don’t seem to remember the lessons of the past.
Each year at the Thanksgiving table we remember - just for a minute and in a story filled with inaccuracies - that there were people here when the Pilgrims travelled to the new world.
That first Thanksgiving when we all shared our first bowl of Chex Mix.
Essay from Dim Sum Thinking Newsletter 191. Read the rest of the Newsletter or subscribe