How fair is my dime, part 2


Continuing my Bayesian experiment with a dime coin. Part one is here.

So, let’s look at the posterior (outcome) probability density function after the next two tosses. First a head:

Two heads, two tails pdfNow we can see the results near the extreme possibilities (ie H= 1, H = 0) being eliminated completely. Interestingly, although, in reality, the evidence (two heads, two tails) points towards a fair coin, ie H = 0.5, the pdf here peaks with H below 0.5, reflecting our earlier ‘estimate’ of a coin biased towards tails.

Now, another tail:

2 heads, 3 tails pdfThe peak is becoming narrower now, with some suggestion that there is a bias to tails, but of course with the coin having only been tossed 5 times, some such “bias” is unavoidable.

Now a head:

Six tosses, three heads, three tails pdfAnd again…

7 tosses pdfAnd again…

pdf after 8 tossesAnd the ninth toss:

pdf after 9 tossesHere the peak is becoming pronounced (and it also looks like the splines used to create the graph are showing signs of Runge’s phenomenon).

And the final, tenth, toss:

pdf after 10 tossesThis does suggest a bias but the pdf is still quite broad – so there is lots of room for further correction. Indeed the thing that strikes me most is how little the suggested bias is after five heads in a row.