A Bayesian Detour 5

You can say, "I don't know anything about A, B, or C!"

You can say that, but it is very often not true. Just a figure of speech. You know some things about the weight of dead meat. No negative numbers. No numbers that are most easily written in scientific notation. I can say that I don't know anything about the weight of a butchered and dressed ox. Except that I do.

It will weigh more than a person. You know that. I know that.

It will weight less than the living beast.

That is information. It is information that everyone has, and it is real. Not everyone knows what a standard living breathing ox of that breed weighs. (I don't.) But we have at least an inkling. We have some general idea, some internal notion. We have some window of weights where we have more and less confidence that we are right.

Between 0 and 10 pounds? No chance. Not at plausible. (If you had to put a probability on that idea of "not at all plausible", what number would you pick?)

Between 0 and a quadrillion pounds? Definitely. Completely certain. (If you had a put a probability on the idea of "total certainty", how would you do that?)

I don't know anything about cows. Or at least, that's what I might say to myself if I didn't feel any incentive to the give the matter conscious deliberation. And yet, as it turns out when I have reason to actually think hard about it for a few moments, it just so happens that I know a few small things about cows. Not much, but a little. Anyone who fills out that form has an infinite range of choices available to write on their card, but there are some numbers that are more reasonable, and some numbers that are less reasonable. And there is, as it happens, some range of reasonableness.

There are ranges broad enough that they are virtually certain, and there are choices (like minus 10 pounds) that are literally impossible.

There is certainty at one end, and impossibility at the other end. That is genuine knowledge. To say it again: this necessarily means that there is some spectrum of belief in-between those two extremes. (What kind of symbolic representation might we want to use for that intermediate uncertainty?)

Mathematicians have a certain aesthetic of absolute certainty. They like their "information", as it were, in the form of definite proof with no wiggle room. But that is not how human knowledge works, a great deal of the time. Once we have finally acknowledged that some forms of knowledge genuinely are fuzzier than others -- that some things we know with great certainty and others we do not -- then we might want some formal framework into which to arrange that continuum of uncertainty.

There is ignorance, yes. There is uncertainty, both Knightian and otherwise. Reading about Galton's numerical analysis, I realized that I have no real idea about what it means for an animal to be "dressed" after it is butchered. I don't think they put a tailored suit on the carcass, but otherwise, not quite sure what that word means in this case. I've never weighed a cow before, or a steer, or an ox, or a butchered ox. Never compared the weight of a living one to a dead one. There is ignorance, yes. There is so much ignorance.

Yet there is also knowledge. There is information.

If someone puts a gun to my head and threatens to shoot me if I guess wrong outside a narrow certain band, then I'm most likely going to get shot. My ignorance is too great to save me. But that does not mean that my ignorance is all-encompassing. I'm going to throw a number out there, and that number will not be negative 10. My ignorance will likely get me killed, but I'm still going to draw on whatever limited resources I have inside this head of mine. Probably not enough. My ignorance is fighting with my information. But that does not mean that that information does not exist. It does.

And we know it exists, because we can see the mean of those answers.

We're back to the WTF question. Here is what is weird about this: My own ignorance is based on my own personal history. Maybe I'd been reading about elephants recently, and the numbers in my head from that book have created a mental anchor -- mixed together with the bajillion other experiences of my life -- and the high anchor from the elephant weight and everything else I've experienced makes me guess too high.

There are many things that we know, that we don't realize that we know. I know maybe five words of French, yet on the rare occasion when I hear people on the train speaking it, I can recognize it immediately. Why? I don't know. Not exactly.

It's the nasal-ness? The up-and-down tones? Definitely part of it, but there have got to be many other reasons for my knowledge here. And this is knowledge. This is information, even if it's not mathematical proof. I've given two conscious reasons why I might recognize this language when I hear it spoken, but there have got to be a hundred more reasons, inside my head, that I don't recognize. We can know some things so deeply that we don't even know that we know them.

And now the Galton gangsters put a gun to your head, and you give your own guess. You are, I presume, as ignorant of the weight of a butchered ox as I am. But supposing that you have some incentive to treat the problem seriously, you're not going to say you just don't know. You're not going to guess zero pounds, nor a million. Your ignorance is vast, but there is information in your head, too. The bajillion little subconscious things are going to form together in your mind, and you're going to provide an answer. The answer will, of course, be wrong because your ignorance is real. Maybe you recently read a book on chimpanzees, and the weights in that book anchor your mind on an answer too low.

Someone like Galton can take all of those guesses and look at them and see that the errors are not strongly correlated with each other. He knows the real weight, and he has the various guesses, and he can express each guess not in terms of its actual number, but in terms of its number minus the true value. And he can write down not the numbers of the guesses, but the errors. A long list of this:


And so on. And this is the key. The errors are not correlated with one another. They appear to be, at first approximation, independent. Just because my personal guess might have been too high did not mean that your person guess would also be too high. My ignorance, and other people's ignorance, did not match each other.

Their ignorance is not related to my own. I am ignorant. You are ignorant. They are ignorant. But the inherent nature of our ignorance somehow stems from different sources. We are ignorant in independent ways, derived from our independent life experiences and weird mental anchors, and that is precisely why adding up all of our errors together cancels them out completely.

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