(...yields a rational number)

But, thank you for your answer...

I thought there would be something simpler because when a and b are natural numbers, the function yields a natural number.

sorry for the late reply. $I$ is defined here: mathworld.wolfram.com/RegularizedBetaFunction.html but there is no expression for $I_0.5$

Oh, I assumed that people get up from the table when they can no longer be head of the table... Sorry.

In fact, this is clearly not right for n=4: heads-heads selects person 0, and tails-tails person 2, but person 1 is selected for either heads-tails or tails-heads. Thus, the probability is not uniform.

Thank you very much for your detailed answer! Am I right that "theta" represents categorical parameter vector observations (i.e., their components are in [0,1])? If the observations are "alpha", the parameter vectors of Dirichlet observations, then the update of v needs to be: v_N[j] = v_0[j] + sum_n [ digamma(sum_i alpha_n[i]) - digamma(alpha_n[j]) ] because that's the expected log of each component value, right? Thanks again. This was very helpful.

...does not list the conjugate prior.