## Cell phone story — part 3

After talking about this problem in class on Monday, I realized that we don’t understand the texting patterns of undergraduates very well. So I’m reluctant to specify an informative prior in this situation for the simple reason that I don’t understand this that well. So I’m going to illustrate using bugs to fit this model using a vague prior.

I have observed the number of text messages for my son for 16 days in this months billing period. Since his monthly allocation is 5000 messages, I am focusing on the event “number of messages in the 30 days exceeds 5000.” Currently my son has had 3314 messages and so I’m interested in computing the predictive probability that the number of messages in the remaining 14 days exceeds 1686.

Here I’ll outline using the openbugs software with the R interface in the BRugs package to fit our model.

I’m assuming that you’re on a Windows machine and have already installed the BRugs package in R.

First we write a script that describes the normal sampling model. Following Kruschke’s Doing Bayesian Data Analysis text, I can enter this model into the R console and write it to a file model.txt.

modelString = ”

model {

for( i in 1:n ){

y[i] ~ dnorm( mu, P )

}

mu ~ dnorm( mu0, P0 )

mu0 <- 100

P0 <- 0.00001

P ~ dgamma(a, b)

a <- 0.001

b <- 0.001

}

”

writeLines(modelString, con=”model.txt”)

# We load in the BRugs package (this includes the openbugs software).

library(BRugs)

# Set up the initial values for and in the MCMC iteration — this puts the values in a file called inits.txt.

bugsInits(list(c(mu = 200, P = 0.05)),

numChains = 1, fileName = “inits.txt”)

# Have openbugs check that the model specification is okay.

modelCheck(“model.txt”)

# Here is our data — n is the number of observations and y is the vector of text message counts.

dataList = list(

n = 16,

y=c(207, 121, 144, 229, 113, 262, 169, 330,

168, 132, 224, 188, 231, 207, 268, 321)

)

# Enter the data into openbugs.

modelData( bugsData( dataList ))

# Compile the model.

modelCompile()

# Read in the initial values for the simulation.

modelInits(“inits.txt”)

# We are going to monitor the mean and the precision.

samplesSet( c(“mu”, “P”))

# We’ll try 10,000 iterations of MCMC.

chainLength = 10000

# Update (actually run the MCMC) — it is very quick.

modelUpdate( chainLength )

# The function samplesSample collects the simulated draws, samplesStats computes summary statistics.

muSample = samplesSample( “mu”)

muSummary = samplesStats( “mu”)

PSample = samplesSample( “P”)

PSummary = samplesStats( “P”)

# From this output, we can compute summaries of any function of the parameters of interest (like the normal standard deviation) and compute the predictive probability of interest.

Let’s focus on the prediction problem. The variable of interest is z, the number of text messages in the 14 remaining days in the month. The sampling model assumes that the number of daily text messages is N(), so the sum of text messages for 14 games is N(.

To simulate a single value of z from the posterior predictive distribution, we (1) simulate a value of from its posterior distribution, and (2) simulate z from a normal distribution using these simulated draws as parameters.