More cognition enhancement in the media

The Daily Mail has had an article on cognition enhancers, with their typical slant: Illegal 'smart drugs' bought online by teenagers before exams could have catastrophic effect on their health. Unsurprisingly taking a lot of stimulants and not sleeping is bad for you. I'm quoted there, of course, promoting responsible use.

It also led to an interview on BBC Oxford, but I can't find it on-line.

To survive one's profession

I discuss the future of youth unemployment in Kvällsposten: Att Överleva Sitt Yrke (in Swedish).

My main point is that demographic change leads to differences in what jobs and careers mean: less transfer of wealth from older generations to younger, companies and even jobs survive shorter than individuals. Hence we need to enable more youth entrepreneurship and flexibility on the job market, or we will end up with a very stiff gerontocracy.

Corrupting the Youth

I am in the Oxford student newspaper: Uni scientist advocates ‘smart drugs’ for students - OK, they sexed up the title a bit, since my long list of ethical and practical caveats would have made a lousy title ("Uni scientist advocates taking smart drugs if you, after studying the evidence and evaluating your ethical stance, use them responsibly with informed consent and an eye towards what kind of social game a university is").

Currently in Lugano, where the enhancer of choice is of course espresso.

The sun rises more surely with Jeffreys than Laplace

The sunrise problem is one of the perennial problems of probability and particularly relevant for the research we do at FHI: how do we estimate a probability for something we have never seen?

The "classic" solution is Laplace's rule of succession that provides us with the answer that if we have seen N sunrises, the probability for another sunrise tomorrow should be (N+1)/(N+2).

This can be calculated this way: Seeing N out of N possible occurrences of an event with true probability p has probability p^N. If we know this has happened, we have a probability distribution for p of the form (N+1)p^N, where the (N+1) terms is a normalization constant. Calculating the expectation for p we get (N+1)/(N+2).

However, this assumes an uniform prior over p: each probability is equally likely (the principle of indifference). While this might seem reasonable, probabilities in the real world tend to either be very small (something almost never happens) or very large (it almost surely happens). Worse, when estimating probabilities we are often interested in order of magnitude rather than absolute values. But a uniform distribution over log p is not uniform over p.

One approach to this is to use an "un-informative" prior, a prior estimate that expresses our uncertainty but is also invariant over the transformations we might think are reasonable for the problem. In this case the Jeffreys prior seems useful. In particular, the version of it (it has a different version for different problems) we want is the one used for estimating a biased coin, J(p) = 1/sqrt(p(1-p)).

Plugging the prior into the previous analysis and using Bayes' rule we get
P(p | having seen N out of N) = (N+1)p^N J(p) / integral₀¹ (N+1)p^N J(p) dp
Unfortunately the integral below the denominator is a messy hypergeometric function. The forms for integer N are simpler (no hypergeometrics) but still unwieldy.

Using numeric integration instead produces the following probability estimates (care must be taken when integrating close to 0 and 1, since J(p) is badly behaved there. Whee!)

The blue curve is using the Jeffreys prior, the red Laplace. So if we believe this prior is better than the uniform one, we will be more confident that the sun will rise tomorrow. Which is a bit surprising, given that the prior actually puts much of its probability mass close to zero. But that part of the prior quickly gets "overruled" by the N observations, amplifying the effect of the other big lump of probability mass near 1.