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Modeling fire alarm counts

Sorry for the long delay since my last post.  It seems that one needs to keep a regular posting pattern, say several days a week. to keep this blog active.

We’re now talking about comparing models by Bayes factors.  To motivate the discussion of plausible models, I found the following website that gives the number of fire calls for each month in Franklinville, NC for the last several years.

Suppose we observe the fire call counts $y_1, ..., y_N$ for $N$ consecutive months.  Here is a general model for these data.

1. $y_1, ..., y_N$ are independent $f(y | \theta)$
2. $\theta$ has a prior $g(\theta)$
Also, suppose we have some prior beliefs about the mean fire count $E(y)$.  We believe that this mean is about 70 and the standard deviation of this guess is 10.
Given this general model structure, we have to think of possible choices for $f$, the sampling density.  We think of the popular distributions, say Poisson, normal, exponential, etc.  Also we should think about different choices for the prior density.   For the prior, there are many possible choices — we typically choose one that can represent my prior information.
Once we decide on several plausible choices of sampling density and prior, then we’ll compare the models by Bayes factors.  To do this, we compute the prior predictive density of the actual data for each possible model.  It is very convenient to perform this calculation for discrete models (where the parameter $\theta$ is discrete) and we’ll first illustrate Bayes factor computations in this special case