Today we talked about using a beta prior to learn about a proportion. Inference about p is done by use of the beta posterior distribution and prediction about future samples is done by means of the predictive distribution.

Here are the R computations for the cell-phone example. I’ll first illustrate inference for the proportion p, and then I’ll illustrate the use of the special function pbetap (in the LearnBayes package) to compute the beta-binomial predictive distribution to learn about the number of successes in a future sample.

> library(LearnBayes)

>

> a=6.8; b=2.5 # parameters of beta prior

> n=24; y=9 # sample size and number of yes’s in sample

>

> a1=a+y; b1=b+n-y # parameters of beta posterior

>

> # I’ll illustrate different types of inferences

>

> # a point estimate is given by the posterior mean

> a1/(a1+b1)

[1] 0.4744745

>

> # or you could find the posterior median

> qbeta(.5,a1,b1)

[1] 0.4739574

>

> # a 90% interval estimate is found by use of the 5th and 95th quantiles

> # of the beta curve

> qbeta(c(.05,.95),a1,b1)

[1] 0.3348724 0.6158472

>

> # we illustrate prediction by use of the function pbetap

> # suppose we take a future sample of 20

> # how many will be driving when using a cell phone?

>

> m=20; ys=0:m

> pred.probs=pbetap(c(a1,b1),m,ys)

>

> # display the predictive probabilities

>

> cbind(ys,pred.probs)

ys pred.probs

[1,] 0 7.443708e-05

[2,] 1 6.444416e-04

[3,] 2 2.897264e-03

[4,] 3 8.968922e-03

[5,] 4 2.139155e-02

[6,] 5 4.170364e-02

[7,] 6 6.884411e-02

[8,] 7 9.841322e-02

[9,] 8 1.236003e-01

[10,] 9 1.376228e-01

[11,] 10 1.365218e-01

[12,] 11 1.208324e-01

[13,] 12 9.524434e-02

[14,] 13 6.650657e-02

[15,] 14 4.075296e-02

[16,] 15 2.159001e-02

[17,] 16 9.665296e-03

[18,] 17 3.527764e-03

[19,] 18 9.889799e-04

[20,] 19 1.901993e-04

[21,] 20 1.891124e-05

>

> # what is the probability that there are at least

> # 10 cell phone drivers in my sample?

>

> sum(pred.probs*(ys>=10))

[1] 0.4958392

>