STAT 330 November 14, 2012

We spent the first part of the period looking at this code example. In this example we are studying the evidence that a coin is fair or not, given that we observed 60 heads and 40 tails. The code illustrates with this simple example the idea of Reversible Jump MCMC, where we propose simultaneously a new model and a value of the parameter (which is 0.5 if the coin is fair, but uniformly distributed on (0,1) if the coin is not fair, in this – highly unrealistic – example). The code allows you to use one of three proposals for the parameter on the unfair case – the exact beta distribution on 60 heads and 40 tails, a normal approximation, and a flat distribution. I pointed out that the first two ought to do pretty well, but that the flat distribution will often propose the parameter out in the tails of the distribution where the posterior probability is low, in which case the proposal is likely to be rejected. We ran the program and found that these predictions were borne out.

We spent the rest of the class looking at various aspects of the Ockham’s razor idea, that one should choose models that are as simple as possible but which still fit the data adequately. Too complex a model is likely to follow “noise” in the data, and too simple will not be adequate to predict the data. We looked at it from the point of view of the idea that a simple hypothesis predicts fewer outcomes than a complex one can. We saw how this worked in the case an alleged planet that had been announced around a pulsar; of proving plagiarism by encoding errors or other unique information into maps, mathematical tables, etc; and how it might be used to detect cheating on multiple choice tests.  We also saw how the hypothesis of copying DNA from ancestors provides evidence for evolution, e.g., pseudogened that used to code for vitamin C production have the same defects in humans and chimpanzees, indicating descent from a common ancestor, and the redundancy of the genetic code provides independent evidence as well (there are 64 combinations of three base pairs in the genetic code, but only 20 amino acids are coded for).

I ended by describing Mercury’s perihelion motion. We will finish this example next time.

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