Bard has given you a biased coin, and you wonder what the probability \(\pi\) of heads is. He replies that he doesn't quite know, but that his friend Sam, who gave it to him, once estimated the probability of heads to be \(\frac{3}{7}\), and that Sam was as certain of that as if he had flipped the coin 21 times.

(a) What are your prior hyperparameters \(a_{0}\) and \(b_{0}\) ?

(b) You get 23 heads and 18 tails. What are the posterior hyperparameters?

(c) You decide to flip the coin a few more times, to get an even sharper estimate on \(\pi\). What are your prior hyperparameters this time? \({ }^{4}\)

(d) You now get 458 heads and 366 tails. What are your posterior hyperparameters now?