Bayesian Thinking

Do graduate students prefer McDonalds?

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Today I did an illustration of discrete Bayes for a proportion.  I’m interested in the proportion p of graduate students who answer “McDonalds” when asked the question “McDonalds, Wendys, or Burger King?”

I believe p can be one of the five values 0.1, 0.2, 0.3, 0.4, 0.5 and I assign the respective prior weights 1, 2, 5, 10, 5.  I define this prior in R:

> p=c(.1,.2,.3,.4,.5)
> prior=c(1,2,5,10,5)
> prior=prior/sum(prior)
> names(prior)=p

> p=c(.1,.2,.3,.4,.5)
> prior=c(1,2,5,10,5)
> prior=prior/sum(prior)
> names(prior)=p

I collected data from my class.  Of the 25 students, 11 responded with “McDonalds”.
I update my probabilities using the function discrete.bayes.  You can read in the function and associated plot and print methods by typing in
source(“http://bayes.bgsu.edu/m6480/R/discrete.bayes.functions.R”)
The updating is done by discrete.bayes.  The arguments are the sampling density dbinom, the prior probabilities defined in prior, the number of yes’s (11) and the sample size (25).
> s=discrete.bayes(dbinom,prior,11,size=25)
I compare the prior and posterior probabilities using two bar graphs.
> par(mfrow=c(2,1))
> barplot(prior,ylim=c(0,.6),xlab=”p”,main=”PRIOR”)
> plot(s,xlab=”p”,main=”POSTERIOR”)
Note that the posterior probs are more precise than the prior probabilities.  I am more confident that the proportion of McDonalds fans is equal to 0.4.
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