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The canonical link for the binomial is the logit. The linear predictor can be anything so it is usable for the probability after the logit transform is used. The case is analogous for the Poisson distribution. But the canonical link for the exponential distribution is the inverse function, so the inverse of the mean is equal to the linear predictor. But this allows the mean to be negative, which is strange because the exponential distribution has the positive line as a domain. Does anyone have an explanation to make sense of this?

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    $\begingroup$ I mean the linear predictor is not usable for the probability itself... $\endgroup$
    – Jan van den Broek
    Commented Jan 11, 2017 at 16:50
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    $\begingroup$ There is no reason why the canonical link should necessarily be a good choice for a GLM and can have akward properties like the one you pointed out. See also e.g. stats.stackexchange.com/questions/40876/… . I believe most people use $\log \lambda = x \beta$. $\endgroup$
    – Björn
    Commented Jan 11, 2017 at 17:26

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