Home > Model checking & comparison > Three Sampling Models for Fire Calls

## Three Sampling Models for Fire Calls

Continuing our example, suppose our prior beliefs about the mean count of fire calls $\theta$ is Gamma(280, 4).  (Essentially this says that our prior guess at $\theta$ is 70 and the prior standard deviation is about 4.2.)  But we’re unsure about the sampling model — it could be (M1) exponential($\theta$), (M2) Poisson($\theta$), or (M3) normal with mean $\theta$ 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 $\theta$ 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 $f(y)$.   (I wrote a short function defining the log posterior of $\theta$ and then used the laplace function in the LearnBayes package to compute $\log f(y)$.)

Here are the results:

Model     $\log f(y)$
————————–
Poisson              -148.0368
exponential        -194.3483
normal(12)        -144.3027
—————————

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

$BF_{NP} = \exp( -144.3037 + 148.0368 ) = 41.8$