So if we attach some probability to the statement “2 + 2 = 4,” then what should the probability be? What you seek to attain in a case like this is good calibration—statements to which you assign “99% probability” come true 99 times out of 100. This is actually a hell of a lot more difficult than you might think. Take a hundred people, and ask each of them to make ten statements of which they are “99% confident.” Of the 1,000 statements, do you think that around 10 will be wrong?
I am not going to discuss the actual experiments that have been done on calibration—you can find them in my book chapter “Cognitive biases potentially affecting judgment of global risks”—because I’ve seen that when I blurt this out to people without proper preparation, they thereafter use it as a Fully General Counterargument, which somehow leaps to mind whenever they have to discount the confidence of someone whose opinion they dislike, and fails to be available when they consider their own opinions. So I try not to talk about the experiments on calibration except as part of a structured presentation of rationality that includes warnings against motivated skepticism.
But the observed calibration of human beings who say they are “99% confident” is not 99% accuracy.
Suppose you say that you’re 99.99% confident that 2 + 2 = 4. Then you have just asserted that you could make 10,000 independent statements, in which you repose equal confidence, and be wrong, on average, around once. Maybe for 2 + 2 = 4 this extraordinary degree of confidence would be possible: “2 + 2 = 4” is extremely simple, and mathematical as well as empirical, and widely believed socially (not with passionate affirmation but just quietly taken for granted). So maybe you really could get up to 99.99% confidence on this one.
I don’t think you could get up to 99.99% confidence for assertions like “53 is a prime number.” Yes, it seems likely, but by the time you tried to set up protocols that would let you assert 10,000 independent statements of this sort—that is, not just a set of statements about prime numbers, but a new protocol each time—you would fail more than once. Peter de Blanc has an amusing anecdote on this point. (I told him not to do it again.)
Yet the map is not the territory: if I say that I am 99% confident that 2 + 2 = 4, it doesn’t mean that I think “2 + 2 = 4” is true to within 99% precision, or that “2 + 2 = 4” is true 99 times out of 100. The proposition in which I repose my confidence is the proposition that “2 + 2 = 4 is always and exactly true,” not the proposition “2 + 2 = 4 is mostly and usually true.”
As for the notion that you could get up to 100% confidence in a mathematical proposition—well, really now! If you say 99.9999% confidence, you’re implying that you could make one million equally fraught statements, one after the other, and be wrong, on average, about once. That’s around a solid year’s worth of talking, if you can make one assertion every 20 seconds and you talk for 16 hours a day.
Assert 99.9999999999% confidence, and you’re taking it up to a trillion. Now you’re going to talk for a hundred human lifetimes, and not be wrong even once?
Assert a confidence of (1 - 1/googolplex) and your ego far exceeds that of mental patients who think they’re God.
And a googolplex is a lot smaller than even relatively small inconceivably huge numbers like 3 ↑↑↑ 3. But even a confidence of (1 - 1∕3 ↑↑↑ 3) isn’t all that much closer to PROBABILITY 1 than being 90% sure of something.
If all else fails, the hypothetical Dark Lords of the Matrix, who are right now tampering with your brain’s credibility assessment of this very sentence, will bar the path and defend us from the scourge of infinite certainty.
Am I absolutely sure of that?
Why, of course not.
As Rafal Smigrodski once said:
I would say you should be able to assign a less than 1 certainty level to the mathematical concepts which are necessary to derive Bayes’s rule itself, and still practically use it. I am not totally sure I have to be always unsure. Maybe I could be legitimately sure about something. But once I assign a probability of 1 to a proposition, I can never undo it. No matter what I see or learn, I have to reject everything that disagrees with the axiom. I don’t like the idea of not being able to change my mind, ever.
*
55
0 And 1 Are Not Probabilities
One, two, and three are all integers, and so is negative four. If you keep counting up, or keep counting down, you’re bound to encounter a whole lot more integers. You will not, however, encounter anything called “positive infinity” or “negative infinity,” so these are not integers.
Positive and negative infinity are not integers, but rather special symbols for talking about the behavior of integers. People sometimes say something like, “5 + infinity = infinity,” because if you start at 5 and keep counting up without ever stopping, you’ll get higher and higher numbers without limit. But it doesn’t follow from this that “infinity - infinity = 5.” You can’t count up from 0 without ever stopping, and then count down without ever stopping, and then find yourself at 5 when you’re done.
From this we can see that infinity is not only not-an-integer, it doesn’t even behave like an integer. If you unwisely try to mix up infinities with integers, you’ll need all sorts of special new inconsistent-seeming behaviors which you don’t need for 1, 2, 3 and other actual integers.
Even though infinity isn’t an integer, you don’t have to worry about being left at a loss for numbers. Although people have seen five sheep, millions of grains of sand, and septillions of atoms, no one has ever counted an infinity of anything. The same with continuous quantities—people have measured dust specks a millimeter across, animals a meter across, cities kilometers across, and galaxies thousands of lightyears across, but no one has ever measured anything an infinity across. In the real world, you don’t need a whole lot of infinity.
(I should note for the more sophisticated readers in the audience that they do not need to write me with elaborate explanations of, say, the difference between ordinal numbers and cardinal numbers. Yes, I possess various advanced set-theoretic definitions of infinity, but I don’t see a good use for them in probability theory. See below.)
In the usual way of writing probabilities, probabilities are between 0 and 1. A coin might have a probability of 0.5 of coming up tails, or the weatherman might assign probability 0.9 to rain tomorrow.
This isn’t the only way of writing probabilities, though. For example, you can transform probabilities into odds via the transformation O = (P∕(1 - P)). So a probability of 50% would go to odds of 0.5/0.5 or 1, usually written 1:1, while a probability of 0.9 would go to odds of 0.9/0.1 or 9, usually written 9:1. To take odds back to probabilities you use P = (O∕(1 + O)), and this is perfectly reversible, so the transformation is an isomorphism—a two-way reversible mapping. Thus, probabilities and odds are isomorphic, and you can use one or the other according to convenience.
For example, it’s more convenient to use odds when you’re doing Bayesian updates. Let’s say that I roll a six-sided die: If any face except 1 comes up, there’s a 10% chance of hearing a bell, but if the face 1 comes up, there’s a 20% chance of hearing the bell. Now I roll the die, and hear a bell. What are the odds that the face showing is 1? Well, the prior odds are 1:5 (corresponding to the real number 1/5 = 0.20) and the likelihood ratio is 0.2:0.1 (corresponding to the real number 2) and I can just multiply these two together to get the posterior odds 2:5 (corresponding to the real number 2/5 or 0.40). Then I convert back into a probability, if I like, and get (0.4∕1.4) = 2∕7 = ~29%.
So odds are more manageable for Bayesian updates—if you use probabilities, you’ve got to deploy Bayes’s Theorem in its complicated version. But probabilities are more convenient for answering questions like “If I roll a six-sided die, what’s the chance of seeing a number from 1 to 4?” You can add up the probabilities of 1/6 for each side and get 4/6, but you can’t add up the odds ratios of 0.2 for each side and get an odds ratio of 0.8.
Why am I saying all this? To show that “od
d ratios” are just as legitimate a way of mapping uncertainties onto real numbers as “probabilities.” Odds ratios are more convenient for some operations, probabilities are more convenient for others. A famous proof called Cox’s Theorem (plus various extensions and refinements thereof) shows that all ways of representing uncertainties that obey some reasonable-sounding constraints, end up isomorphic to each other.
Why does it matter that odds ratios are just as legitimate as probabilities? Probabilities as ordinarily written are between 0 and 1, and both 0 and 1 look like they ought to be readily reachable quantities—it’s easy to see 1 zebra or 0 unicorns. But when you transform probabilities onto odds ratios, 0 goes to 0, but 1 goes to positive infinity. Now absolute truth doesn’t look like it should be so easy to reach.
A representation that makes it even simpler to do Bayesian updates is the log odds—this is how E. T. Jaynes recommended thinking about probabilities. For example, let’s say that the prior probability of a proposition is 0.0001—this corresponds to a log odds of around -40 decibels. Then you see evidence that seems 100 times more likely if the proposition is true than if it is false. This is 20 decibels of evidence. So the posterior odds are around -40 dB + 20 dB = -20 dB, that is, the posterior probability is ~0.01.
When you transform probabilities to log odds, 0 goes onto negative infinity and 1 goes onto positive infinity. Now both infinite certainty and infinite improbability seem a bit more out-of-reach.
In probabilities, 0.9999 and 0.99999 seem to be only 0.00009 apart, so that 0.502 is much further away from 0.503 than 0.9999 is from 0.99999. To get to probability 1 from probability 0.99999, it seems like you should need to travel a distance of merely 0.00001.
But when you transform to odds ratios, 0.502 and 0.503 go to 1.008 and 1.012, and 0.9999 and 0.99999 go to 9,999 and 99,999. And when you transform to log odds, 0.502 and 0.503 go to 0.03 decibels and 0.05 decibels, but 0.9999 and 0.99999 go to 40 decibels and 50 decibels.
When you work in log odds, the distance between any two degrees of uncertainty equals the amount of evidence you would need to go from one to the other. That is, the log odds gives us a natural measure of spacing among degrees of confidence.
Using the log odds exposes the fact that reaching infinite certainty requires infinitely strong evidence, just as infinite absurdity requires infinitely strong counterevidence.
Furthermore, all sorts of standard theorems in probability have special cases if you try to plug 1s or 0s into them—like what happens if you try to do a Bayesian update on an observation to which you assigned probability 0.
So I propose that it makes sense to say that 1 and 0 are not in the probabilities; just as negative and positive infinity, which do not obey the field axioms, are not in the real numbers.
The main reason this would upset probability theorists is that we would need to rederive theorems previously obtained by assuming that we can marginalize over a joint probability by adding up all the pieces and having them sum to 1.
However, in the real world, when you roll a die, it doesn’t literally have infinite certainty of coming up some number between 1 and 6. The die might land on its edge; or get struck by a meteor; or the Dark Lords of the Matrix might reach in and write “37” on one side.
If you made a magical symbol to stand for “all possibilities I haven’t considered,” then you could marginalize over the events including this magical symbol, and arrive at a magical symbol “T” that stands for infinite certainty.
But I would rather ask whether there’s some way to derive a theorem without using magic symbols with special behaviors. That would be more elegant. Just as there are mathematicians who refuse to believe in the law of the excluded middle or infinite sets, I would like to be a probability theorist who doesn’t believe in absolute certainty.
*
56
Your Rationality Is My Business
Some responses to Lotteries: A Waste of Hope chided me for daring to criticize others’ decisions; if someone else chooses to buy lottery tickets, who am I to disagree? This is a special case of a more general question: What business is it of mine, if someone else chooses to believe what is pleasant rather than what is true? Can’t we each choose for ourselves whether to care about the truth?
An obvious snappy comeback is: “Why do you care whether I care whether someone else cares about the truth?” It is somewhat inconsistent for your utility function to contain a negative term for anyone else’s utility function having a term for someone else’s utility function. But that is only a snappy comeback, not an answer.
So here then is my answer: I believe that it is right and proper for me, as a human being, to have an interest in the future, and what human civilization becomes in the future. One of those interests is the human pursuit of truth, which has strengthened slowly over the generations (for there was not always Science). I wish to strengthen that pursuit further, in this generation. That is a wish of mine, for the Future. For we are all of us players upon that vast gameboard, whether we accept the responsibility or not.
And that makes your rationality my business.
Is this a dangerous idea? Yes, and not just pleasantly edgy “dangerous.” People have been burned to death because some priest decided that they didn’t think the way they should. Deciding to burn people to death because they “don’t think properly”—that’s a revolting kind of reasoning, isn’t it? You wouldn’t want people to think that way, why, it’s disgusting. People who think like that, well, we’ll have to do something about them . . .
I agree! Here’s my proposal: Let’s argue against bad ideas but not set their bearers on fire.
The syllogism we desire to avoid runs: “I think Susie said a bad thing, therefore, Susie should be set on fire.” Some try to avoid the syllogism by labeling it improper to think that Susie said a bad thing. No one should judge anyone, ever; anyone who judges is committing a terrible sin, and should be publicly pilloried for it.
As for myself, I deny the therefore. My syllogism runs, “I think Susie said something wrong, therefore, I will argue against what she said, but I will not set her on fire, or try to stop her from talking by violence or regulation . . .”
We are all of us players upon that vast gameboard; and one of my interests for the Future is to make the game fair. The counterintuitive idea underlying science is that factual disagreements should be fought out with experiments and mathematics, not violence and edicts. This incredible notion can be extended beyond science, to a fair fight for the whole Future. You should have to win by convincing people, and should not be allowed to burn them. This is one of the principles of Rationality, to which I have pledged my allegiance.
People who advocate relativism or selfishness do not appear to me to be truly relativistic or selfish. If they were really relativistic, they would not judge. If they were really selfish, they would get on with making money instead of arguing passionately with others. Rather, they have chosen the side of Relativism, whose goal upon that vast gameboard is to prevent the players—all the players—from making certain kinds of judgments. Or they have chosen the side of Selfishness, whose goal is to make all players selfish. And then they play the game, fairly or unfairly according to their wisdom.
If there are any true Relativists or Selfishes, we do not hear them—they remain silent, non-players.
I cannot help but care how you think, because—as I cannot help but see the universe—each time a human being turns away from the truth, the unfolding story of humankind becomes a little darker. In many cases, it is a small darkness only. (Someone doesn’t always end up getting hurt.) Lying to yourself, in the privacy of your own thoughts, does not shadow humanity’s history so much as telling public lies or setting people on fire. Yet there is a part of me which cannot help but mourn. And so long as I don’t try to set you on fire—only argue with your ideas—I believe that it is right and proper to me, as a human, that I care about my fellow humans. That, also, is a position I defend into the Future.
*
Part F
Politics and Rationality
57
Politics is the Mind-Killer
People go funny in the head when talking about politics. The evolutionary reasons for this are so obvious as to be worth belaboring: In the ancestral environment, politics was a matter of life and death. And sex, and wealth, and allies, and reputation . . . When, today, you get into an argument about whether “we” ought to raise the minimum wage, you’re executing adaptations for an ancestral environment where being on the wrong side of the argument could get you killed. Being on the right side of the argument could let you kill your hated rival!
If you want to make a point about science, or rationality, then my advice is to not choose a domain from contemporary politics if you can possibly avoid it. If your point is inherently about politics, then talk about Louis XVI during the French Revolution. Politics is an important domain to which we should individually apply our rationality—but it’s a terrible domain in which to learn rationality, or discuss rationality, unless all the discussants are already rational.
Politics is an extension of war by other means. Arguments are soldiers. Once you know which side you’re on, you must support all arguments of that side, and attack all arguments that appear to favor the enemy side; otherwise it’s like stabbing your soldiers in the back—providing aid and comfort to the enemy. People who would be level-headed about evenhandedly weighing all sides of an issue in their professional life as scientists, can suddenly turn into slogan-chanting zombies when there’s a Blue or Green position on an issue.
In Artificial Intelligence, and particularly in the domain of nonmonotonic reasoning, there’s a standard problem: “All Quakers are pacifists. All Republicans are not pacifists. Nixon is a Quaker and a Republican. Is Nixon a pacifist?”
Rationality- From AI to Zombies Page 24
