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## Discrete Bayes

One good way of introducing Bayesian inference is by the use of discrete priors. I recently wrote a simple generic R function that does discrete Bayes for arbitrary choice of a prior and sampling density. I’ll illustrate this function here and in the next posting.

Suppose I’m interested in learning about the rate parameter of an exponential density. We observe a sample from the density defined by

where we are interested in learning about the rate parameter .

In R, I read in the function “discrete.bayes” and a few associated methods by typing

> source(“http://bayes.bgsu.edu/m6480/R/discrete.bayes.functions.R”)

I define my prior. I believe the values .05, .10, …, 0.50 are all plausible rate values and I assign each the same prior probability. In the following R code, prior is a vector of probabilities where I’ve named the probabilities by the values of the rate.

> rate=seq(.05,.50,by=.05)

> prior=rep(.1,10)

> names(prior)=rate

> prior

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1

Now I enter in the observations:

> y=c(6.2, 5.0, 1.2, 10.2, 5.9, 2.3, 1.1, 19.0, 4.2, 27.5)

I update my probabilities by the function discrete.bayes — the arguments are the sampling density (dexp), the prior vector, and the data vector.

> s=discrete.bayes(dexp,prior,y)

To display, graph, and summarize these probabilities, I use the generic functions “print”, “plot”, and “summary”.

Here are the posterior probabilities.

> print(s)

0.05 0.1 0.15 0.2 0.25 0.3

2.643537e-02 4.353606e-01 4.037620e-01 1.153126e-01 1.727192e-02 1.719954e-03

0.35 0.4 0.45 0.5

1.292256e-04 7.900085e-06 4.125920e-07 1.903091e-08

To display the probabilities, use “plot”.

> plot(s,xlab=”RATE”,ylab=”PROBABILITY”)

To find a 90% probability estimate for the rate, use summary:

> summary(s,coverage=0.90)

$coverage

[1] 0.9544352

$set

[1] 0.10 0.15 0.20

The probability the rate is in the set {0.10, 0.15, 0.20} is 0.95.

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