So going over the Hastie Tibshirani paper on GAM - it points to equation 11 as the exponential family density - but with two parameters - theta (natural parameter) and phi (scale).

https://projecteuclid.org/euclid.ss/1177013604 $$ p(x;\theta,\phi) = \exp \left\{ \frac{[\theta \cdot x−A(\theta)]} {a(\phi)} +c(x,\phi) \right\} $$ where $\cdot$ is a dot product and $c$ and $a$ are functions.

Whereas I am more familiar with the location only representation - see for example Definition of exponential family

$$ p(x;\theta) = h(x) \exp( \theta \cdot T(x)−A(\theta)) $$ where $\cdot$ is a dot product. What is going on? its likely the difference between density and distribution with the phi parameters defining the normalizing constant.

Would like some details or a reference would be most appreciated. Observations:

  1. The $c(x,\phi)$ is $c(x)$ or basically $h(x) = \exp(c(x))$ and $a(\phi) =1 $. No idea why x became T(x)
  2. Note the derivative of the cumulant $A(\theta)$ in each equation is equal to the expected value of x and T(x) respectively. WHich indicates the two are not equivalent.


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