Grappling With The Hinge Of History Part 2: Bayesian Dilemma

TLDR: I lean towards using Laplace’s rule of succession to start quantifying whether it’s the hinge of history.^[1] That means since I’m focused on the hinge of the future, the initial odds this century is the most influential future century would be ½.^[2][3]

Will MacAskill thinks there’s a 0.1 to 1% chance the current century is the hinge of history. Toby Ord thinks there’s a ⅙ chance there’ll be an existential catastrophe by 2120.^,

Yet Ord’s biggest disagreement with MacAskill isn’t related to climate change, nuclear war, pandemics or AI. It’s about how to set a Bayesian prior.^[4]

MacAskill’s Prior

The first question MacAskill asks to come up with his Bayesian prior (prior) is how long will anyone be alive? MacAskill uses the self-sampling assumption to estimate this number. He states, “the prior probability of us living in the most influential century, conditional on Earth-originating civilization lasting for n centuries, is 1/n.” 

MacAskill then calculates n. MacAskill estimates that 1 trillion years is an early estimate for the end of time because that’s when stars could stop forming.^[5] MacAskill estimates there’s a 0.01% chance anyone survives that long, but he doesn’t say why. .01% of 1 trillion years equals 1 million centuries, so MacAskill starts with a prior of 1 in 1 million that the current century is the most influential century. ^[6]

MacAskill thinks this starting point is low enough that it’s extremely unlikely someone lives during the hinge of history. He says he’d be suspicious of any calculations that suggest otherwise.

Civil internet discourse ensued.

Is the most influential century more likely to happen earlier?

Ord observes that MacAskill is trying to set his prior by assuming there’s an equal chance over time that an event, the hinge of history, happens in any century. Ord believes that if an event has an equal chance of occurring in any century over time, it’s more likely to happen in an earlier century.

I think Ord’s point makes sense. Imagine hypothetical event X has a 1% chance of occurring each century. After 100 centuries, there’s a 63.4% chance it would’ve happened.

If you want to understand how it’s 63.4% and you know multiplication, here’s an explanation of how that math works. (If you already understand this point or trust my math, scroll down to the next section.) First, imagine you were flipping a coin. For each toss, the odds of heads is 50% and the odds of tails is 50%. 

What are the odds you toss heads twice in a row? It would be .5 * .5. That equals .25, 25%.

What are the odds you toss heads three times in a row? It would be .5 * .5 * .5. That’s .5^3, which equals .125, 12.5%.

Now, what if you wanted to figure out the odds you’ll get tails at least once if you toss a coin three times?

We just calculated the odds of getting heads three times. So every other scenario such as getting tails three times in a row or getting heads twice and then one tails, would include getting at least one tails. 

That means you could subtract the odds of getting heads three times from 1. 1 - .125 = .875. That means there’s an 87.5% chance of getting at least one tails when you flip a coin three times.

The same logic applies to any other event with different probabilities. 

Imagine a random number generator that randomly returns a whole number from 1 through 100. Each time you get a random number, there’s a 1% chance the number is 1, a 1% chance the number is 2, a 1% chance the number is 3, etc. That means there’s a 99% chance the number is not 1 each time a random number is generated.

So what are the odds you don’t get 1 the first two times you run the random number generator? It’s .99 * .99. That equals .9801, 98.01%. 

And what are the odds you get 1, one of the first two times you run the random number generator? It’s 1 - .9801. That equals .0199, 1.99%.

The same logic from the coin flip example applies to the random number generator. The only difference is that there’s a 1% chance each random number generated is a 1 while there’s a 50% chance a coin is heads.

So the odds you don’t get 1 the first one hundred times you run the random number generator are .99 ^ 100. That’s .3660323413, 36.6%. 

That means the odds you get at least one 1 the first one hundred times you run the random number generator are 1 - .3660323413 = .6339676587. That’s 63.4%. 

Now compare two of the questions. What are the odds of getting a 1 during 100 runs of a random number generator from 1-100? And what are the odds hypothetical event X with a 1% chance of happening in a century occurs within 100 centuries? They’re essentially the same question.

And humans, which Ord seems to define as “anatomically modern” Homo sapiens, have been around for 200,000 years per fossil records.^[7]

200,000 years is 2000 centuries. (200000 / 100 = 2000)

So if there was a 1% chance an event took place per century over 2000 centuries, there’s a 99.999999981% chance that event happened. (1 - .99^2000 = .99999999981. Multiply that by 100 to get 99.999999981%.) Even if it was only a .01% chance per century over 2000 centuries, there’s a 18.12774347% chance the event happened. (1 - .9999^2000 = .1812774347. 1812774347 * 100 = 18.12774347%.) Just over 18% may sound small, but if there were 1 million centuries, as MacAskill estimates, the first 2000 centuries would only be the first .2% of centuries. (200 / 1000000 = .002. .002 * 100 = .2%.)

That’s why I’m persuaded by Ord’s point that if you assume an event can only happen once and it has a constant probability of occurring at any period in history, it would probably take place earlier in time.

Laplace’s Rule of Succession

So what should the prior be to determine that any century is the most influential? This is difficult because it doesn’t make sense to assume there’s a 1% chance, or whatever percent chance, that a century is the most influential. The percentage is unclear. 

Ord says Laplace’s rule of succession is the most well-known way to determine the odds an event with an unknown probability will occur.^[8] 

Laplace’s rule says that when one doesn’t know the odds an event will occur and one knows the event could both occur or not occur, take the fraction of times the event has occurred so far and add one to the numerator and two to the denominator.

I don’t understand the mathematical proof for Laplace’s rule, but I understand the intuition. (This article helped me understand it. I essentially explain it below.)

Imagine the app developer who released the random number generator has followed up on their massive success with a coin flipper app. However, you’re warned this virtual coin might not be a fair coin. 

You’re given two other pieces of information. First, it won’t land on heads every time and it won’t land on tails every time. It still could land on heads or tails every time, but one. Second, like a regular coin flip, the outcome of each virtual flip is independent. That means it’s not affected by the results of prior flips.

So what are the odds that the first virtual flip will be heads? There’s no reason you should have any intuition about whether it should be heads or tails. 

That means it would be logical to assume there’s a 50% chance it’ll be heads. Notice you just applied Laplace’s rule. 50% is ½. The event of heads had happened 0 times and you added 1 to it. All events, heads and tails, had happened 0 times and you added 2 to it.

To continue applying Laplace’s rule, keep on recording the events. If the first flip is heads add 1 to the numerator and denominator. So if that happened, you should now guess the odds of heads are ⅔. If the second flip is tails, add one to the denominator. That’s 2/4. So the odds of heads for the third flip are 50% again.

So what should the prior be that the current century is the hinge of history per Laplace’s rule? The odds would be ½ before the first century. If it wasn’t the hinge of history, the odds entering the second century would be ⅓. Since humans have been around 2000 centuries, add the remaining 1999 centuries to get the prior the current century is the hinge of history. That’s 1/2002.^[9] 

Jeffrey’s Prior

But Ord clarifies that he didn’t use Laplace’s rule. He explains that he doesn’t think the initial odds a century is the most influential should be constant. 

This means he doesn’t think the unfair virtual coin would be an accurate model for determining the prior. When predicting the results of the virtual coin flip, it was assumed that the probability the virtual coin would land on heads or tails was the same on each individual flip. We just didn’t know it. So we tried to predict that probability based on the data we had and updated our estimate of those probabilities after each flip. 

Ord says that the initial odds the first century is the most influential shouldn't be the same as the initial odds the second century is the most influential etc.  

He says that he used a bucket-shaped Jeffrey’s prior. I’m assuming it would look kind of like the below picture:

This might be more of a U than a bucket

But besides the Laplacean prior, there’s no such thing as an uninformed prior. Every other prior implicitly updates the Laplacean prior of ½.

For example, let’s say someone was trying to predict the odds that an American who turns 95 dies over the next year. One reasonable way to set a prior would be to use data indicating the percentage of 95-year-olds who die before they turn 96, 25.87%. Then someone could update their guess based on whether the 95-year-old American is sick or there’s a pandemic, etc. 

But data from thousands of Americans were used to inform the “prior” that there’s a 25.87% chance that someone will die when they’re 95. The first being, which we’ll pretend is human for this hypothetical, wouldn’t be able to use any historical mortality data. When they were born they’d have no idea what the odds are that they’d make it to 1-year-old. And they wouldn’t have any reason to know they’d be more likely to die when they’re 95 than when they’re 18.

So I think using a Laplacean prior to predict the odds it’s the hinge of history definitely seems fair. That prior only incorporates the data that (anatomically modern) humans have lived for 2000 centuries.

I don’t mean to suggest that all we know is that there have been 2000 centuries so far. A lot of things have happened in those 2000 centuries. But it seems bold for Ord to claim that some centuries have different prior probabilities to be the most influential. I wish he’d further explained his reasoning. (Ord did add that he used (= Bets(0.5, 0.5) for his Jeffrey’s prior. I don’t know what that means.)

The Jeffrey’s prior led Ord to cut his prior that this is the most influential century, in half to 1/4004.

Setting priors based on population size

Ord explains that he thinks it makes sense to set the prior based on the number of people alive rather than the number of centuries. He points out that 1/20, 5%, of all the people who have ever existed are alive today. This means the prior this is the most influential time period would be 1/20 with Laplace’s rule and 1/40 with his Jeffrey’s prior.^[10] 

This seems intuitive to me. MacAskill agreed too. I presume the more people alive during a time period, the more influential it will be. (Granted, this rate may not increase linearly. People can make an impact by preventing others from doing something. For example, a U.S. president could make a sizable impact by vetoing bills passed by a legislature controlled by the opposing party.)

Ord adds that people are also living longer than they ever have before. Apparently, the years lived by people alive today make up 15% of the years all humans have lived. I don’t know what those numbers are projected to be by the end of the century. 

Did the most influential century already happen?

As I said in part 1, I’m focused on improving the future rather than knowing the hinge of history for the sake of it. But I did find it interesting that Ord said that using his model there was “a 99.999% chance or something” that the most influential time in history already happened. However, he doesn’t think it’s happened yet.^[11]

I wish Ord elaborated on that point too. How could he come up with that number? To do so, wouldn’t someone have to ask something like what are the odds an event with a 1% chance, or any concrete probability, happens over 2,000 occurrences? Isn’t Laplace’s rule or a Jeffrey’s prior used because we don’t know the odds an event will occur?

I’m also skeptical that the prior should be a big factor for determining the most influential century. Presumably, if I thoroughly studied human history, I’d update the prior a ton.

For now, I lean towards using Laplace’s rule to start asking whether it’s the most influential century. And since I’m focused on the most influential future century, I’m starting from scratch. 

So my prior this is the most influential future century is ½. I’ll look into updating the prior in my next post.

1. ^{^}

   I chose to use MacAskill’s original phrasing of the hinge of history question as “Are we living in the most influential time ever?” instead of “Am I one of the most influential people ever?” That’s because I’m assuming that predicting whether an individual will be influential would largely be based on their personal characteristics.

2. ^{^}

   Ord defines existential catastrophe as an event that causes “the destruction of humanity’s longterm potential.” I wouldn’t take this too literally. For example, if I procrastinate for 1 minute, have I reduced humanity’s potential? Based on reading Ord’s book, I think Ord used the term to represent one significant event that prevents society from progressing past its current level or leads it to get worse. I’m not sure what progress or worse means to him. So an existential catastrophe and the hinge of history aren’t necessarily the same thing. For the purposes of this article, I don’t think that matters. I consider the hinge of history and an existential catastrophe to both be events that don’t have a known initial probability. As a result, I prefer to set both events' prior using Laplace’s rule of succession. I’ll explain Laplace’s rule later in the post.

3. ^{^}

   And Holden Karnofsky thinks the odds we’re in the most important century are 15-30%.Karnofsky seems to consider the most important century to be the hinge of history.

^{^}

MacAskill, Ord and Karnofsky agree on using Bayesian statistics, but it’s debatable whether Bayesian or frequentist statistics should be used to determine probabilities. My impression is that it would be harder to get started determining whether a century is the most influential using a frequentist approach. The most influential time would need to be defined in a way that one could gather data to objectively answer the question. Bayesianism appears to be more popular too, although I haven't found precise data confirming that.

^{^}

 It’s estimated to take 1-100 trillion years for stars to stop forming normally.

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MacAskill edits his original post to say that he’s focusing on “the claim that we are at the most influential century AND that we have an enormous future ahead of us.”  I’m not sure if he was referring to his entire claim about the hinge of history or the part he’s writing about priors. Since he refers to extinction risk many times in his subsequent post, I’ll assume he’s only referring to priors. I’m not clear why he thinks it makes sense to assume that we have an enormous future ahead of us to help form an accurate prior.

^{^}

There were fossils found from approximately 315,000 years ago that seem to be considered Homo sapiens that are not anatomically modern. This is summarized here and here. There is also DNA evidence, which suggests modern humans originated 260,000-350,000 years ago.

^{^}

Ord uses the alternate terminology Laplace’s law of succession. Laplace’s rule of succession seems to be the more commonly used terminology.

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In Ord’s reply to MacAskill, he says he’ll say 1/2000 instead of 1/2002. He says the plus two is there because they’ve arbitrarily chosen to determine intervals of importance by century. I get Ord’s point that time is continuous and each time is the time for an infinitesimally short amount of time. But, as long as the question discussed is the most influential century, I think it still makes sense to say 1/2002. As I covered in part 1, I agree choosing to evaluate centuries is arbitrary.

^{^}

I presume he used 1/20 to simplify his numbers. If 7.8 billion people are alive and 117 billion people have been alive, 6.67% of all people who have ever lived are alive today.

^{^}

Ord’s statement confuses me because it implies he has a model that’s specifically about the hinge of history rather than an existential catastrophe. My best guess is that he’s saying summing the prior probabilities of previous centuries from his model would indicate that the hinge of history or an existential catastrophe should’ve already happened. And based on his updates to the priors for past centuries, he thinks it’s unlikely either event has already happened.

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