Somehow I ended up on the curious site LessWrong.com, directed by somebody with the nickname Yud (from Eliezer Yudkowsky). Odd isn’t in it. Investigating it led me to Reddit and this explanation of the strange views at LessWrong.
Yud’s main issue in terms of his ‘personal philosophy’ is a slavish adherence to the idea of ‘Bayesianism’ and his view of ‘rationality’, even when it does not actually make sense. Yud rejects the idea of anything having absolutely 0% or absolutely 100% probablity, which leads to some very odd opinions and conclusions. In one of Yud’s own essays, he points out that if you’re mugged and asked for $50 or the mugger will detonate a nuclear bomb in the city (but he doesn’t have a gun pointed at you), then you should give the mugger the money – after all, there’s a nonzero chance of the city blowing up!
Now whether this Yud fellow holds these views, or whether this is a fair summary of his mugger problem, it’s good enough for us to demonstrate two important truths: that all probability is conditional, and that probability is not decision. (This is not Pascal’s Mugger, which we’ll do another day.)
Would you give the mugger the fifty bucks? If you’re a guy who makes a living writing a blog to drum up business, you wouldn’t have it. You’d have to negotiate and see if he’d take twenty. Or ten. (Note to certain readers: I initially wrote this post before Thanksgiving.)
But suppose instead you were a government employee with a guaranteed pension. You’d probably laugh in the mugger’s face.
First, our proposition of interest runs something like, Y = “The mugger has a nuclear device and it will work and he will use it to murder millions if I don’t give him fifty bucks.” The probability of that is what we want. What is it?
It doesn’t exist. There is no Pr(Y), i.e. there is no probability of Y. Yet it seems to us there is and that it is low, even negligible, and even zero. Why is that? Because we are each of us supplying premises, evidence and observations, from which we can form a probability. That is each of us is arguing (with each of our own selves) something like this:
E = “This guy is a nut. Nuts don’t have access to nuclear devices. How could he have got one? I suppose it’s possible, but there are so many safeguards and security precautions that I can’t see how he could. Besides, nutty people say things like this. And, anyway, even if he had it, would he really kill millions for a mere fifty bucks? At best, he’s got some wires strung together that he’s calling a bomb.”
And from those not-too-coherent but genuine premises, we form Pr(Y | E) = minuscule or, for most of us, Pr(Y | E) = 0; i.e. the probability that Y is so given our internal argument is tiny, negligible, close to zero as you like. Maybe even zero depending on the precise form of the evidence.
It’s not that we form this E is some systematic way. It, or some thing like it, appears in chaotic fashion to each of us. Those that consider Y, at any rate. Not everybody will understand Y. That the mugger made himself plain is another premise we assume. Many people, in Manhattan, would hear the mugger say, “Do you have a minute for…” and pass right on by.
Importantly, if Pr(Y | E) = minuscule, minuscule is not a number. Our E is far too fuzzy for us to deduce an quantity for the probability. If one came to a number, it is because, and only because, one has included in E a way for a deduction from E to a quantity for Y. E.g. E includes “It’s impossible for a mugger to have such a device”, then Pr(Y | E) = 0.
Now it’s clear enough there will be two many camps, based on two general forms of evidence. There will be a group which says Pr(Y | E) = minuscule, and other which says Pr(Y | E) = 0. The second predominates. For those who hold Pr(Y | E) = 0, it’s also clear which decision is best. Walk on; ignore the mugger; or consider reporting him to the police so that he might be tossed into the loony bin.
What about the Pr(Y | E) = minuscule crowd? What decision should they make? First, and by now obvious, the decision is not the probability. No part of the decision formed any part of the evidence for Y. And even if it had, by accident or through faulty reason, put the decision and its consequences as part of E, this would merely be a mistake.
Understand, the probability is not the thing. If you judge Pr(Y | E) = minuscule and it turns out the mugger was lying or telling the truth, “Pr(Y | E)” was still true, or correct.
What decision to make then depends on some rule which takes the probability as (if you like) input. This isn’t the space to investigate all the different proposed rules (minimax, maximin, expected value, and others), except to say that these rules all take consideration of what happens once you make a decision (hand over your money or walk on) and what happens with the mugger (he has or hasn’t a bomb).
None of those pairs (there are four) will be quantifiable, though many would fake it by, for instance, assigning so many dollars or “utiles” (fictional quantums of feeling), and then fixing a number on the probability and solving the equation given by the rule. That works when infinities don’t crop up, as they sometimes do.
Anyway, there are surely awful consequences if you decide to walk away and the mugger has (and uses) the bomb. But too awful to contemplate, for most of us, especially on the street in front of what might be a nut. Best most of us would do in the “heat” of that moment is to say, fleetingly, to ourselves, “Gosh.” The allure of continuing is much easier to contemplate, and would overwhelm the vague harm of millions. So even with the decision rules, given crude quantifications, we would make the “optimal” decision to pass on by.
Like I said above, we’ll do Pascal’s Mugging another day (soon).
Update Just in time for Christmas, a book which explains all of probability and statistics: Uncertainty: The Soul of Modeling, Probability & Statistics.