Expressing individual distributions in the general exponential form

In Section 3.1.3 we introduced the idea of expressing distributions in a general form for exponential distributions:

Forms for a, b and c were given for the Bernoulli, binomial and Poisson distributions. We now show how these forms are obtained from the distribution densities.

The Bernoulli distribution

is, by logging and then exponentiating the right-hand side of this equation, rearranged in the general exponential form as f (y) = exp[y log(x/(1 - ¡)) + log(1 - ¡)].

Thus, 0 = log(x/(1 - ¡)) and we obtain the logit as the canonical link function. The mean, can then be expressed as the inverse of the logit function, ¡x = exp(0)/(1 + exp(0)) = (1 + exp(-0))-1. Writing the distribution in terms of 0 we obtain f (y) = exp{y0 + log[1 - exp(0)/(1 + exp(0))]} = exp{y0 - log[1 + exp(0)]}.

Therefore, b(0) = log(1 + exp(0)), a = 1 and c(y) = 1.

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