# Where can I read about the theory behind GLMs in their most general form? [duplicate]

I was following along with the MIT Opencourseware "Statistics for Applications" and their analysis of GLMs only covers discussion of GLMs whose dependent variable $$y$$ is distributed according to a distribution in the canonical exponential family (i.e. of the form below): $$P(y\ |\ \theta) = h(y)\exp(\theta^Ty - A(\theta))$$ If this is the case, then one can easily determine that $$\mu = \mathbb{E}[Y] = A'(\theta)$$, and since $$\mu$$ is related to the linear predictor via the link function $$g$$, we have $$X^T\beta = g(A'(\theta)),\qquad \theta = (g\circ A')^{-1}(X^T\beta)$$

However, I'm curious about the more general case where the distribution belongs to the overdispersed exponential family: $$P(\mathbf y\ |\ \theta,\tau) = h(y,\tau)\exp\left(\frac{\mathbf b(\mathbf \theta)^T\mathbf{T}(\mathbf y) - A(\theta)}{d(\tau)}\right)$$

Using the identity $$\mathbb{E}[\nabla_\theta\ell] = 0$$ where $$\ell$$ is the log-likelihood function I was able to arrive at $$[D_\theta \mathbf b]^T\mathbb{E}[\mathbf T(\mathbf y)] = \nabla_\theta A$$ but there are two problems with this. For one, the dimension of $$\mathbf b$$ might not equal the dimension of $$\theta$$, and even if it did this would not guarantee the matrix $$D_\theta\mathbf b$$ (the Jacobian of $$\mathbf b$$) is invertible. Second, even if it were invertible, we would only arrive at $$\mathbb{E}[\mathbf T(\mathbf y)] = ([D_\theta \mathbf b]^T)^{-1}\nabla_\theta A$$ which doesn't give us any direct correspondence between the parameter $$\theta$$ and the linear predictor $$\mathbf X^T\beta$$, since the link function links $$\mathbb{E}[\mathbf y]$$ to $$\mathbf X^T\beta$$, not $$\mathbb{E}[\mathbf T(\mathbf y)]$$.

So, how are GLMs of this form implemented? Where can I go to read more about how this works?