ABSTRACT
The density g(ir) can be taken as the prior distribution of the param eter 7r, from which the posterior will be derived. If each fi is a beta of typel, beta(ai, /?*), i = 1, . . . , m, we have a finite mixture of betas, which can present one, or several modes. For Bernoulli sampling, the posterior density p(7r|n, r) can be obtained very simply by using the conjugate property of each individual beta, with, however, changes in the posterior weights. We have
In general, we can prove that
above case, is
When considering only two densities, the posterior weights in the mixture can also be determined from the Bayes factor in favor of f \, which is the ratio of the posterior odds to the prior odds. We have:
. Since we also have
Figure 1 shows the mixture of 3 betas,
Mixtures of beta distributions could be very useful in many respects. First, since they can display one or several modes, they can be used to approximate complex densities. Second, they can adequately rep resent the weighted average of several sources of information. Third, with Bernoulli or Pascal sampling, they are easy to use, and provide solutions in closed forms, as shown above.
