## Three Sampling Models for Fire Calls

Continuing our example, suppose our prior beliefs about the mean count of fire calls is Gamma(280, 4). (Essentially this says that our prior guess at is 70 and the prior standard deviation is about 4.2.) But we’re unsure about the sampling model — it could be (M1) exponential(), (M2) Poisson(), or (M3) normal with mean and standard deviation 12.

To get some sense about the best sampling model, I’ve plotted a histogram of the fire call counts below. I’ve overlaid fitted exponential, Poisson, and normal distributions where I estimate by the sample mean.

I think it is pretty clear from the graph that the exponential model is a poor fit. The fits of the Poisson and normal (sd = 12) models are similar, although I’d give a slight preference to the normal model.

For each model, I compute the logarithm of the predictive probability . (I wrote a short function defining the log posterior of and then used the laplace function in the LearnBayes package to compute .)

Here are the results:

Model

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Poisson -148.0368

exponential -194.3483

normal(12) -144.3027

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The exponential model is a clear loser and the Poisson and normal(12) models are close. The Bayes factor in support of the normal(12) model is