Bayesian thinking
Exercise
Explain and reconcile the following three results.
(1) Researchers Griffiths (Brown) and Tenenbaum (MIT) gave “nuggets of information to each of the participants in their study …, 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.” (“Bayes Rules,” The Economist, January 7, 2006, pp. 70-71).
(2) Kahneman and Tversky found, when asked to estimate the percentage of African states at the United Nations, respondents estimated on average 25 and 45% when, before being asked, they extracted from a lottery a number which could be either 10 or 65, respectively (Frank, 1992, 277).
(3) A sample of students divides by half in preferring one of two apartments, A and B, characterized on the basis of distance and price. When a third apartment, C, which being more distant and costing more than B, it is therefore “dominated” by B, is also offered, more students prefer now B to A. It seems inconsistent because the presence of an irrelevant option seems to be modifying the preferences, the price the students are willing to pay for shorter distance (Frank, 1992, 279-281
Analysis
Apparently, the human mind is a good Bayesian calculator, well equipped to maximize the information in any small piece of data, to the point that this mental process may be tricked with artificial settings such as the one in the pure lottery. Something similar may be going on in the third example: students may be using the new information on the availability of C to revise their estimate of their own cost of distance. Some of these revisions may be “irrational” as in the lottery case, but hardly in a natural environment.
An open question is how “artificial” is today’s world, meaning by artificial the propensity to create “lottery” situations, in which events are really independent. Independent events are possibly very few in nature. There is scope for systematic failure, however, and the persistence of superstition is probably rooted in overstretching our Bayesian machinery.
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