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Two probabilities
Consider the following statements:
1. The result of this coin flip is heads.
2. There is life on Mars.
3. The millionth digit of pi is odd.
What is the probability of each statement?
A frequentist might say, "P1 = 0.5. P2 is either epsilon or 1-epsilon, we don't know which. P3 is either 0 or 1, we don't know which."
A Bayesian might reply, "P1 = P2 = P3 = 0.5. By the way, there's no such thing as a probability of exactly 0 or 1."
Which is right? As with many such long-unresolved debates, the problem is that two different concepts are being labeled with the word 'probability'. Let's separate them and replace P with:
F = the fraction of possible worlds in which a statement is true. F can be exactly 0 or 1.
B = the Bayesian probability that a statement is true. B cannot be exactly 0 or 1.
Clearly there must be a relationship between the two concepts, or the confusion wouldn't have arisen in the first place, and there is: apart from both obeying various laws of probability, in the case where we know F but don't know which world we are in, B = F. That's what's going on in case 1. In the other cases, we know F != 0.5, but our ignorance of its actual value makes it reasonable to assign B = 0.5.
When does the difference matter?
Suppose I offer to bet my $200 the millionth digit of pi is odd, versus your $100 that it's even. With B3 = 0.5, that looks like a good bet from your viewpoint. But you also know F3 = either 0 or 1. You can also infer that I wouldn't have offered that bet unless I knew F3 = 1, from which inference you are likely to update your B3 to more than 2/3, and decline.
On a larger scale, suppose we search Mars thoroughly enough to be confident there is no life there. Now we know F2 = epsilon. Our Bayesian estimate of the probability of life on Europa will also decline toward 0.
Once we understand F and B are different functions, there is no contradiction.
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Demands for Particular Proof: Appendices
Appendices to: You're Entitled to Arguments, But Not (That Particular) Proof
(The main article was getting long, so I decided to move the appendices to a separate article which wouldn't be promoted, thus minimizing the size of the article landing in a promoted-article-only-reader's feed.)
A. The absence of unobtainable proof is not even weak evidence of absence.
The wise will already know that absence of evidence actually is evidence of absence; and they may ask, "Since a time-lapse video record of apes evolving into humans would, in fact, be strong evidence in favor of the theory of evolution, is it not mandated by the laws of probability theory that the absence of this videotape constitute some degree of evidence against the theory of evolution?"
(Before you reject that proposition out of hand for containing the substring "evidence against the theory of evolution", bear in mind that grownups understand that evidence accumulates. You don't get to pick out just one piece of evidence and ignore all the rest; true hypotheses can easily generate a minority of weak pieces of evidence against themselves; conceding one point of evidence does not mean conceding the debate; and people who try to act as if it does are nitwits. Also there are probably no creationists reading this blog.)
The laws of probability theory do mandate that if P(H|E) > P(H), then P(H|~E) < P(H). So - even if absence of proof is by no means proof of absence, and even if we reject the philosophy that absence of a particular proof means you get to discard all the other arguments about evidence and priors - must we not at least concede that absence of proof is necessarily evidence of absence, even though it may be very weak evidence?
Advancing Certainty
Related: Horrible LHC Inconsistency, The Proper Use of Humility
Overconfidence, I've noticed, is a big fear around these parts. Well, it is a known human bias, after all, and therefore something to be guarded against. But I am going to argue that, at least in aspiring-rationalist circles, people are too afraid of overconfidence, to the point of overcorrecting -- which, not surprisingly, causes problems. (Some may detect implications here for the long-standing Inside View vs. Outside View debate.)
Here's Eliezer, voicing the typical worry:
[I]f you asked me whether I could make one million statements of authority equal to "The Large Hadron Collider will not destroy the world", and be wrong, on average, around once, then I would have to say no.
I now suspect that misleading imagery may be at work here. A million statements -- that sounds like a lot, doesn't it? If you made one such pronouncement every ten seconds, a million of them would require you to spend months doing nothing but pontificating, with no eating, sleeping, or bathroom breaks. Boy, that would be tiring, wouldn't it? At some point, surely, your exhausted brain would slip up and make an error. In fact, it would surely make more than one -- in which case, poof!, there goes your calibration.
No wonder, then, that people claim that we humans can't possibly hope to attain such levels of certainty. Look, they say, at all those times in the past when people -- even famous scientists! -- said they were 99.999% sure of something, and they turned out to be wrong. My own adolescent self would have assigned high confidence to the truth of Christianity; so where do I get the temerity, now, to say that the probability of this is 1-over-oogles-and-googols?
Drawing Two Aces
Suppose I have a deck of four cards: The ace of spades, the ace of hearts, and two others (say, 2C and 2D).
You draw two cards at random.
Scenario 1: I ask you "Do you have the ace of spades?" You say "Yes." Then the probability that you are holding both aces is 1/3: There are three equiprobable arrangements of cards you could be holding that contain AS, and one of these is AS+AH.
Scenario 2: I ask you "Do you have an ace?" You respond "Yes." The probability you hold both aces is 1/5: There are five arrangements of cards you could be holding (all except 2C+2D) and only one of those arrangements is AS+AH.
Now suppose I ask you "Do you have an ace?"
You say "Yes."
I then say to you: "Choose one of the aces you're holding at random (so if you have only one, pick that one). Is it the ace of spades?"
You reply "Yes."
What is the probability that you hold two aces?
Argument 1: I now know that you are holding at least one ace and that one of the aces you hold is the ace of spades, which is just the same state of knowledge that I obtained in Scenario 1. Therefore the answer must be 1/3.
Argument 2: In Scenario 2, I know that I can hypothetically ask you to choose an ace you hold, and you must hypothetically answer that you chose either the ace of spades or the ace of hearts. My posterior probability that you hold two aces should be the same either way. The expectation of my future probability must equal my current probability: If I expect to change my mind later, I should just give in and change my mind now. Therefore the answer must be 1/5.
Naturally I know which argument is correct. Do you?
The Amanda Knox Test: How an Hour on the Internet Beats a Year in the Courtroom
Note: The quantitative elements of this post have now been revised significantly.
Followup to: You Be the Jury: Survey on a Current Event
All three of them clearly killed her. The jury clearly believed so as well which strengthens my argument. They spent months examining the case, so the idea that a few minutes of internet research makes [other commenters] certain they're wrong seems laughable
- lordweiner27, commenting on my previous post
The short answer: it's very much like how a few minutes of philosophical reflection trump a few millennia of human cultural tradition.
Wielding the Sword of Bayes -- or for that matter the Razor of Occam -- requires courage and a certain kind of ruthlessness. You have to be willing to cut your way through vast quantities of noise and focus in like a laser on the signal.
But the tools of rationality are extremely powerful if you know how to use them.
Rationality is not easy for humans. Our brains were optimized to arrive at correct conclusions about the world only insofar as that was a necessary byproduct of being optimized to pass the genetic material that made them on to the next generation. If you've been reading Less Wrong for any significant length of time, you probably know this by now. In fact, around here this is almost a banality -- a cached thought. "We get it," you may be tempted to say. "So stop signaling your tribal allegiance to this website and move on to some new, nontrivial meta-insight."
But this is one of those things that truly do bear repeating, over and over again, almost at every opportunity. You really can't hear it enough. It has consequences, you see. The most important of which is: if you only do what feels epistemically "natural" all the time, you're going to be, well, wrong. And probably not just "sooner or later", either. Chances are, you're going to be wrong quite a lot.
Probability Space & Aumann Agreement
The first part of this post describes a way of interpreting the basic mathematics of Bayesianism. Eliezer already presented one such view at http://lesswrong.com/lw/hk/priors_as_mathematical_objects/, but I want to present another one that has been useful to me, and also show how this view is related to the standard formalism of probability theory and Bayesian updating, namely the probability space.
The second part of this post will build upon the first, and try to explain the math behind Aumann's agreement theorem. Hal Finney had suggested this earlier, and I'm taking on the task now because I recently went through the exercise of learning it, and could use a check of my understanding. The last part will give some of my current thoughts on Aumann agreement.
You Be the Jury: Survey on a Current Event
As many of you probably know, in an Italian court early last weekend, two young students, Amanda Knox and Raffaele Sollecito, were convicted of killing another young student, Meredith Kercher, in a horrific way in November of 2007. (A third person, Rudy Guede, was convicted earlier.)
If you aren't familiar with the case, don't go reading about it just yet. Hang on for just a moment.
If you are familiar, that's fine too. This post is addressed to readers of all levels of acquaintance with the story.
What everyone should know right away is that the verdict has been extremely controversial. Strong feelings have emerged, even involving national tensions (Knox is American, Sollecito Italian, and Kercher British, and the crime and trial took place in Italy). The circumstances of the crime involve sex. In short, the potential for serious rationality failures in coming to an opinion on a case like this is enormous.
Now, as it happens, I myself have an opinion. A rather strong one, in fact. Strong enough that I caught myself thinking that this case -- given all the controversy surrounding it -- might serve as a decent litmus test in judging the rationality skills of other people. Like religion, or evolution -- except less clichéd (and cached) and more down-and-dirty.
Of course, thoughts like that can be dangerous, as I quickly recognized. The danger of in-group affective spirals looms large. So before writing up that Less Wrong post adding my-opinion-on-the-guilt-or-innocence-of-Amanda-Knox-and-Raffaele-Sollecito to the List of Things Every Rational Person Must Believe, I decided it might be useful to find out what conclusion(s) other aspiring rationalists would (or have) come to (without knowing my opinion).
So that's what this post is: a survey/experiment, with fairly specific yet flexible instructions (which differ slightly depending on how much you know about the case already).
Frequentist Statistics are Frequently Subjective
Andrew Gelman recently responded to a commenter on the Yudkowsky/Gelman diavlog; the commenter complained that Bayesian statistics were too subjective and lacked rigor. I shall explain why this is unbelievably ironic, but first, the comment itself:
However, the fundamental belief of the Bayesian interpretation, that all probabilities are subjective, is problematic -- for its lack of rigor... One of the features of frequentist statistics is the ease of testability. Consider a binomial variable, like the flip of a fair coin. I can calculate that the probability of getting seven heads in ten flips is 11.71875%... At some point a departure from the predicted value may appear, and frequentist statistics give objective confidence intervals that can precisely quantify the degree to which the coin departs from fairness...
Gelman's first response is "Bayesian probabilities don't have to be subjective." Not sure I can back him on that; probability is ignorance and ignorance is a state of mind (although indeed, some Bayesian probabilities can correspond very directly to observable frequencies in repeatable experiments).
My own response is that frequentist statistics are far more subjective than Bayesian likelihood ratios. Exhibit One is the notion of "statistical significance" (which is what the above comment is actually talking about, although "confidence intervals" have almost the same problem). Steven Goodman offers a nicely illustrated example: Suppose we have at hand a coin, which may be fair (the "null hypothesis") or perhaps biased in some direction. So lo and behold, I flip the coin six times, and I get the result TTTTTH. Is this result statistically significant, and if so, what is the p-value - that is, the probability of obtaining a result at least this extreme?
Well, that depends. Was I planning to flip the coin six times, and count the number of tails? Or was I planning to flip the coin until it came up heads, and count the number of trials? In the first case, the probability of getting "five tails or more" from a fair coin is 11%, while in the second case, the probability of a fair coin requiring "at least five tails before seeing one heads" is 3%.
Whereas a Bayesian looks at the experimental result and says, "I can now calculate the likelihood ratio (evidential flow) between all hypotheses under consideration. Since your state of mind doesn't affect the coin in any way - doesn't change the probability of a fair coin or biased coin producing this exact data - there's no way your private, unobservable state of mind can affect my interpretation of your experimental results."
Why (and why not) Bayesian Updating?
the use of Bayesian belief updating with expected utility maximization may be just an approximation that is only relevant in special situations which meet certain independence assumptions around the agent's actions.
For those who aren't sure of the need for an updateless decision theory, the paper Revisiting Savage in a conditional world by Paolo Ghirardato might help convince you. (Although that's probably not the intention of the author!) The paper gives a set of 7 axioms, based on Savage's axioms, which is necessary and sufficient for an agent's preferences in a dynamic decision problem to be represented as expected utility maximization with Bayesian belief updating. This helps us see in exactly which situations Bayesian updating works and why. (In many other axiomatizations of decision theory, the updating part is left out, and only expected utility maximization is derived in a static setting.)
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