(This article was originally published at Statistical Modeling, Causal Inference, and Social Science, and syndicated at StatsBlogs.)
Deborah Mayo points us to a post by Stephen Senn discussing various aspects of induction and statistics, including the famous example of estimating the probability the sun will rise tomorrow. Senn correctly slams a journalistic account of the math problem:
The canonical example is to imagine that a precocious newborn observes his first sunset, and wonders whether the sun will rise again or not. He assigns equal prior probabilities to both possible outcomes, and represents this by placing one white and one black marble into a bag. The following day, when the sun rises, the child places another white marble in the bag. The probability that a marble plucked randomly from the bag will be white (ie, the child’s degree of belief in future sunrises) has thus gone from a half to two-thirds. After sunrise the next day, the child adds another white marble, and the probability (and thus the degree of belief) goes from two-thirds to three-quarters. And so on. Gradually, the initial belief that the sun is just as likely as not to rise each morning is modified to become a near-certainty that the sun will always rise.
[The above quote is not by Senn; it’s a quote of something he disagrees with!]
Canonical and wrong. X and I discuss this problem in section 3 of our article on the history of anti-Bayesianism (see also rejoinder to discussion here). We write:
The big, big problem with the Pr(sunrise tomorrow | sunrise in the past) argument is not in the prior but in the likelihood, which assumes a constant probability and independent events. Why should anyone believe that? Why does it make sense to model a series of astronomical events as though they were spins of a roulette wheel in Vegas? Why does stationarity apply to this series? That’s not frequentist, it is not Bayesian, it’s just dumb. Or, to put it more charitably, it is a plain vanilla default model that we should use only if we are ready to abandon it on the slightest pretext.
Strain at the gnat that is the prior and swallow the ungainly camel that is the iid likelihood. Senn’s discussion is good in that he keeps his eye on the ball knits his row straight without getting distracted by stray bits of yarn.
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