Here's a puzzle based on something I used to be confused about:

It is known that utility functions are equivalent (i.e. produce the same preferences over actions) up to a positive affine transformation: u'(x) = au(x) + b where a is positive.

Suppose I have u(vanilla) = 3, u(chocolate) = 8. I prefer an action that yields a 50% chance of chocolate over an action that yields a 100% chance of vanilla, because 0.5(8) > 1.0(3).

Under the positive affine transformation a = 1, b = 4; we get that u'(vanilla) = 7 and u'(chocolate) = 12. Therefore I now prefer the action that yields a 100% chance of vanilla, because 1.0(7) > 0.5(12).

How to resolve the contradiction?

I'll admit it: I am confused about genetics and heritability. Not about the results of the various twin studies - Scott summarises them as "~50% of the variation is heritable and ~50% is due to non-shared environment", which seems generally correct.

But I am confused about what this means in practice, due to arguments like "contacts are very important for business success, rich people get much more contacts than poor people, yet business success is strongly correlated with genetic parent wealth" and such. Assuming that genetics strongly determines... most stuff... goes against so many things we know or think we know about how the world works. And by "we" I mean lots of different people with lots of different political views - genetic determinism means, for instance, that current variations in regulation and taxes are pretty unimportant for individual outcomes.

Now, there are many caveats about the genetic results, particularly that they measure the variance of a factor rather than its absolute importance (and hence you get results like variation in nutrition being almost invisible as an explanation for variation in height), but it's still hard to figure out what this all means.

Then we have Scott's latest post, which points out that "non-shared environment" is not the same as "nurture", since it includes, for instance, dumb luck.

However, "heritable" is not the same as as "nature", either. For instance, sexism and racial prejudices, if they are widespread, come under the "heritable" effects rather than the "environment" ones. And then it gets even more confusing.

Widespread prejudice is not "environment". Rarer prejudice is.

For instance, imagine that we lived in a very sexist society where women were not allowed to work at all. Then there would be an extremely high, almost perfect, correlation between "having a Y chromosome" and "having a job". But this would obviously be susceptible to a cultural fix.

Obviously racial effects can have the same effect. It covers anything visible. So a high heritability is compatible with genetics being a cause of competence, and/or prejudice against visible genetic characteristics being important ("Our results indicate that we either live in a meritocracy or a hive of prejudice!").

Note that as prejudices get less widespread, they move from showing up on the genetic variation, to showing up in the environmental variation side. So widespread prejudices create a "nature" effect, rarer ones create a "nurture" effect. Evenly reducing the magnitude of a prejudice, however, doesn't change the side it will show up on.

Positional genetic goods: Beauty... and IQ?

Let's zoom in on one of those visible genetic characteristics: beauty. As Robin Hanson is fond of pointing out, beautiful people are more successful, and are judged as more competent and cooperative than they actually are. Therefore if we have a gene that increases both beauty and IQ, we would expect it's impact on success to be high. In the presence of such a gene, the correlation between IQ and success would be higher than it should objectively be. This suggest a (small) note of caution on the "mutation load" hypotheses; if reducing mutation load increases factors such as beauty, then we would expect increased success without necessarily increased competence.

But is it possible that IQ itself is in part a positional good? Consider that success doesn't just depend on competence, but on social skills, ability to present yourself well in an interview, and how managers and peers judge you. If IQ affects or covaries with one or another of those skills, then we would be overemphasising the importance of IQ in competence. Thus attempts to genetically boost IQ could give less impact than expected. The person whose genome was changed would benefit, but at the (partial) expense of everyone else.

Do people know of experiments (or planned experiments) that disentangle these issues?

This went over well in the xkcd logic puzzle forum (my hand was not removed), so I thought I'd try it here.  It came to me in a dream, so by solving it you may be helping to summon an elder god or something.

In Defense of Objective Bayesianism by Jon Williamson was mentioned recently in a post by lukeprog as the sort of book that should be being read by people on Less Wrong. Now, I have been reading it, and found some of it quite bizarre. This point in particular seems obviously false. If it’s just me, I’ll be glad to be enlightened as to what was meant. If collectively we don’t understand, that’d be pretty strong evidence that we should read more academic Bayesian stuff.

Williamson advocates use of the Maximum Entropy Principle. In short, you should take account of the limits placed on your probability by the empirical evidence, and then choose a probability distribution closest to uniform that satisfies those constraints.

So, if asked to assign a probability to an arbitrary A, you’d say p = 0.5. But if you were given evidence in the form of some constraints on p, say that p ≥ 0.8, you’d set p = 0.8, as that was the new entropy-maximising level. Constraints are restricted to Affine constraints. I found this somewhat counter-intuitive already, but I do follow what he means.

But now for the confusing bit. I quote directly;

“Suppose A is ‘Peterson is a Swede’, B is ‘Peterson is a Norwegian’, C is ‘Peterson is a Scandinavian’, and ε is ‘80% of all Scandinavians are Swedes’. Initially, the agent sets P(A) = 0.2, P(B) = 0.8, P(C) = 1 P(ε) = 0.2, P(A & ε) = P(B & ε) = 0.1. All these degrees of belief satisfy the norms of subjectivism. Updating by maxent on learning ε, the agent believes Peterson is a Swede to degree 0.8, which seems quite right. On the other hand, updating by conditionalizing on ε leads to a degree of belief of 0.5 that Peterson is a Swede, which is quite wrong. Thus, we see that maxent is to be preferred to conditionalization in this kind of example because the conditionalization update does not satisfy the new constraints X’, while the maxent update does.”

p80, 2010 edition. Note that this example is actually from Bacchus et al (1990), but Williamson quotes approvingly.

His calculation for the Bayesian update is correct; you do get 0.5. What’s more, this seems to be intuitively the right answer; the update has caused you to ‘zoom in’ on the probability mass assigned to ε, while maintaining relative proportions inside it.

As far as I can see, you get 0.8 only if we assume that Peterson is a randomly chosen Scandinavian. But if that were true, the prior given is bizarre. If he was a randomly chosen individual, the prior should have been something like P(A & ε) = 0.16 P(B & ε) = 0.04 The only way I can make sense of the prior is if constraints simply “don’t apply” until they have p=1.

Can anyone explain the reasoning behind a posterior probability of 0.8?