Each coin in a collection of bent coins has a different probability of coming up heads when tossed; the probability for each coin i s drawn h m a Beta distribution. Each coin is tossed several times with outcome recorded as I if heads, 0 if tails. Because the lighting is poor, the coin tosser sometimes records the outcome correctly (with probability p, a fixed known quantity)

and sometimes records the opposite outcome (with probability 1 - y ) .

(c)

14.5*

Define the variables needed to specify a hierarchical modcl for this situation, including the recorded outcome of a single toss, the true outcome, and the probability of heads for a single coin. Use subscripts appropriately.

Draw a diagram showing how the variables are rebated in the style of Figure 14.1 (b).

Write out all the conditional distributions needed to complete the model.

Derive an expression for the posterior probability that a particular toss was actually a head given that it was recorded as a head, conditional on the probability of heads for that coin.

Derive an expression for the probability that the coin is recorded as a head given the parameter value for the probability of heads for that coin.

Derive an expression for the posterior distribution of the parameter for the probability of heads for a particular coin, given the Beta prior parameters and the information that it was recorded as having two heads and two tails out of four tosses. (It may involve an integral that you cannot do.)