I was looking at the Poisson regression. Here I find that the canonical parameter is taken to be the logarithm of the rate of the Poisson. I was wondering if there is any reason for considering this as the canonical parameter, rather than the rate of the poisson itself (with the restriction that the parameter is positive).
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1 Answer
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The definition of the canonical (or natural) parameter in the exponential family / generalized linear models is a precise mathematical one. It's not necessarily what might seem "natural" to casual inspection (though it often makes sense).
In particular, for models of the form
$$f(y | \theta, \phi) = h(y,\phi) \exp{\left(\frac{b(\theta)T(y) - A(\theta)}{d(\phi)} \right)}. \,$$
if the parameterization is chosen such that $b(\theta)=\theta$, then $\theta$ is the natural, or canonical parameter. When this paramaterization is used, there are some nice consequences which make it attractive (the natural parameter space is always convex, for example).
Further details are here:
