One way to extend the use of conjugate priors is by the use of discrete mixtures. Here is a simple example. Suppose you have a coin that you believe may be fair or biased towards heads. If p represents the probability of flipping heads, then suppose your prior is

g(p) = 0.5 beta(p, 20, 20) + 0.5 beta(p, 30, 10)

Suppose that we flip the coin 30 and get 20 heads. It can be shown that the posterior density can also be represented by a mixture of beta distributions.

I have written a short R function pbetamix that computes the posterior distribution when a proportion has a prior that is a mixture of beta distributions.

The matrix bpar contains the beta parameters where each row gives the beta parameters for each component. The vector prob gives the prior probabilities of the components. The vector data contains the number of successes and failures. Here are the values of these components for the example.

> bpar=rbind(c(20,20),c(30,10))

> prob=c(.5,.5)

> data=c(20,10)

We use the function pbetamix with the inputs prob, bpar, and data. The output is the posterior probabilities of the components and the beta parameters of each component.

> pbetamix(prob,bpar,data)

$probs

[1] 0.3457597 0.6542403

$betapar

[,1] [,2]

[1,] 40 30

[2,] 50 20

The posterior density is

g(p | data) = 0.346 beta(p, 40, 30) + 0.654 beta(p, 50, 20)

We plot the prior and posterior densities

> plot(p,.5*dbeta(p,30,30)+.5*dbeta(p,30,10),type=”l”,ylim=c(0,5),col=”red”,lwd=2)

> lines(p,.346*dbeta(p,40,30)+.654*dbeta(p,50,20),lwd=2,col=”blue”)

> text(locator(n=1),”PRIOR”)

> text(locator(n=1),”POSTERIOR”)

Here the data (20 heads and 10 tails) is more supportive of the belief that the coin is biased and the posterior places more of its mass where p > .5.