God Plays Dice

An Intuitive Explanation of Bayesian Reasoning, by Eliezer Yudkowsky (of Overcoming Bias fame).

A sequel to this is A Technical Explanation of Technical Explanation [sic].

A list of logical fallacies. (Via Terence Tao's homepage.)

Most of these are extra-mathematical fallacies. For example, in mathematics we don't often see a relation "X causes Y", although it is quite common in ordinary discourse. Even in probability and statistics, we don't see causation nearly as often as correlation. However, they may be familiar to mathematicians as "false methods of proof". For example, what this list calls appeal to force is the well-known "proof by intimidation" -- the Important Person at the front of the room says it's true, so it must be!

A lot of these fallacies are essentially statistical in nature, as any reasoning about the real world must be. In mathematics we either know things or we do not; we don't attach probabilities to our knowledge. (However, we can attach probabilities to other people's knowledge -- or to our own extra-mathematical knowledge -- and then reason about those probabilities. This is the basis of Bayesian reasoning.) Many others are fallacies that exploit the ambiguitiy of natural language. Perhaps the power of mathematics is that it allows us to know things surely, which can never happen in the Real World. But on the flip side, mathematicians know fewer things than other people, because we insist on such certainty in our knowledge.

A Bayesian explanation of how to determine the gender of a person on the street (from observable cues), by Meep.

It's rather similar to Bayesian spam filtering (Paul Graham, see also here). The major difference is that one can generally assume that most e-mail is spam, whereas one cannot assume that most people are of one or the other of the two canonical genders.

In the spam filtering case, though, it doesn't seem that the prior probability that a message is spam matters; Graham claims that most e-mails are either very likely or very unlikely to be spam. But there are probably more words in an e-mail than there are easily observable cues to a person's gender; it seems much more likely to get, say, that a person has a 60% probability of being male than that an e-mail has a 60% probability of being spam.

Also, it's a lot easier to collect the necessary for spam filtering than for gender determination.

Robin Hanson critiques the Doomsday Argument. This is an argument on the lifespan of the human species, which begins from the following principle: there is nothing special about present-day humans. "Therefore" we can consider the number of humans who have lived so far as a fraction of the number of humans who will ever live; the probability that this is between p and q is q-p. I put "therefore" in quotes because the implication is tempting, but one could equally well conclude that the amount of time there have been humans, as a fraction of the amount of time there will ever be humans, has this same distribution. (Indeed, I've heard both versions of the argument.) The first version of this argument says, for example, that the probability that there will be sixty billion more humans is at least one-half; the second says that the probability that we as a species will survive for another two hundred thousand years or so is at least one-half. (I'm assuming there have been sixty billion people who've ever lived and that our species is 200,000 years old.)

And indeed there are other classes of beings that you can use as the reference class here. Living things, for example. Or vertebrates, or living cells, or humans, or even such classes as "humans who haved lived after the year X", which get kind of ridiculous. That last one is particularly prone to abuse, as we can simultaneously say that humanity has a fifty percent chance of surviving past 2114 (if we take X = 1900) and past 3014 (if we take X = 1000).

The name "Doomsday argument" is rather misleading, too. "Doomsday" is usually seen as a bad thing. But what comes after humanity might be the "posthumans" that the people who believe in a technological singularity talk about; is that really doom? Hanson gives a quantitative version of this where there are several "toy universes".

I've talked before about how I'm not entirely comfortable with the "Copernican principle" from which this is derived. For some reason I am much more uncomfortable with this than I would be with the equivalent line of reasoning applied to non-human objects. If I had an urn containing balls labeled from 1 to N, and I didn't know N, and I reached in and grabbed a ball marked 100, I'd say in a heartbeat that the urn probably contained around 200 balls. But the difference is that in the Doomsday Argument we don't even know what the urn is.

The Doomsday argument supposedly only is provisional, until such time as we have better knowledge on how long societies tend to last. This is in my mind one of the most useful reasons for trying to find extraterrestrial intelligence; the knowledge that they do exist (or even a thorough search which doesn't turn up anything) would give us substantial information about how long we might expect to last.

When I studied biochemistry I thought something similar. Essentially all known life forms on Earth have similar biochemistry, because we all evolved from the same ancestors. So an introductory biochemistry class essentially consists of the memorization of those mechanisms. What I would have wanted to see is, say, a dozen or so independently evolved biochemistries, and then see which features of our own biochemistry are just accidents of evolution and which are essential to having complex, self-replicating systems.