The Economist writes: "The key to successful Bayesian reasoning is not in having an extensive, unbiased sample, which is the eternal worry of frequentists, but rather in having an appropriate “prior”, as it is known to the cognoscenti. This prior is an assumption about the way the world works—in essence, a hypothesis about reality—that can be expressed as a mathematical probability distribution of the frequency with which events of a particular magnitude happen.

The best known of these probability distributions is the “normal”, or Gaussian distribution. But there are also the Poisson distribution, the Erlang distribution, the power-law distribution and many even weirder ones that are not the consequence of simple mathematical equations (or, at least, of equations that mathematicians regard as simple).

With the correct prior, even a single piece of data can be used to make meaningful Bayesian predictions. <> Dr Griffiths and Dr Tenenbaum conducted their experiment by giving individual nuggets of information to each of the participants in their study (of which they had, in an ironically frequentist way of doing things, a total of 350), and asking them to draw a general conclusion. For example, many of the participants were told the amount of money that a film had supposedly earned since its release, and asked to estimate what its total “gross” would be, even though they were not told for how long it had been on release so far.

Besides the returns on films, the participants were asked about things as diverse as the number of lines in a poem (given how far into the poem a single line is), the time it takes to bake a cake (given how long it has already been in the oven), and the total length of the term that would be served by an American congressman (given how long he has already been in the House of Representatives). All of these things have well-established probability distributions, and all of them, together with three other items on the list—an individual's lifespan given his current age, the run-time of a film, and the amount of time spent on hold in a telephone queuing system—were predicted accurately by the participants from lone pieces of data."

Here is the paper (pdf). Though they write that "people’s judgments were close to the optimal predictions produced by our Bayesian model across a wide range of settings" it's not so clear how close those predictions really were (see page 24). But the paper is definitely worth reading.

Addendum: Don't miss the comment by Andrew Gelman