I made my one trip to the United States in November 2009 – four days in Washington DC (three working, one extra day’s holiday). It was a fantastic experience and I hope to go again, soon.

I brought back a dime (10 cent coin) from that trip and tonight decided to test it for its fairness as a tossing coin (really, I am working through an example from Data Analysis: A Bayesian Tutorial, and as I write this I do not know what the answer will be.)

I tossed the coin ten times and (taking FDR’s bust as the head, h, and liberty, peace and victory as the tails, t), I got this sequence thththhhhh. So 70% of the tosses resulted in a head – too many to suggest the coin is fair? Let’s see.

I am going to use this formula, based on Bayes’s theorem:

In other words the probability density function (pdf) for bias (H = 0 is a double tailed coin, H = 0.5 is a perfectly fair coin, H = 1 is a double headed coin) is proportional to the products of the observed data and the initially guessed (prior) pdf for H (in all cases I means the other, initial, conditions in the universe).

I begin with stipulating I know nothing about the coin’s bias, in other words set the prior pdf to 1 for all possible values of H. After two tosses we know that the probability of H = 1 is zero and similarly the probability of H = 0 is also zero. But what of the other values for H.

Well, but we ignore that just to look at the relative values.

The data pdf is governed by the binomal distribution where is the total number of tosses and is the number of heads returned.

So for H = 0.1 we get , H = 0.2 , H = 0.3 , H = 0.4 , H = 0.5 . The results are obviously symmetrical about 0.5, and as the prior pdf is 1 for all values of H, we can say that the most likely view is that the coin is fair but we have a high degree of uncertainty (see graph)

So, now we toss an additional tail and take the new pdf (as in the graph) as our prior – what do we get now.

Well for H = 0.1 we now get for the data which we multiply by our new prior of 0.09 to get 0.00729. For H = 0.2 this becomes (of course the actual numbers mean little here, we are merely tracing the shape of the graph). In contrast H = 0.8 would now give . The new graph has the shape shown below:

This coin may be biased but there is still a lot of uncertainty here…

As it’s now after 12.30 in the morning here the rest will have to wait…

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