# Saddlepoint approximation for Exponential family

I read the following in a book:

The saddlepoint approximation of an exponential family density function is $$\tilde P(y;\mu,\phi) = \frac{1}{\sqrt{2\pi \phi V(y)}}exp(-\frac{d(y, \mu)}{2\phi})$$

Where $$d(y, \mu)$$ is the unit deviance; $$V(y) = \frac{d\mu}{d\theta}$$ evaluated at $$\mu=y$$.

I managed to see this for individual distributions (e.g. Poisson) but not for the general form. How do I differentiate the unit deviance twice and get that it's equal to $$V(y)$$?

We want to approximate the $$b(y, \phi)$$ in the following representation of an expo-family pdf: $$\mathcal{P}(y;\mu,\phi) = b(y,\phi)e^{-\frac{d(y,\mu)}{2\phi}}$$ Where the b-function is the regular $$c(y,\phi)$$ of the expo-family + absorbing the $$t(y,y)$$ part of the unit deviance $$d(y,\mu)$$, and we do this using Laplace's method / Saddle point approximation.

So - $$f(y) = e^{-\frac{d(y,\mu)}{2\phi}}$$, i.e. the function without the "bounding constant" (b-function).

$$h(y) = \ln f(y)=-\frac{d(y,\mu)}{2\phi} = -2\frac{t(y,y)-t(y,\mu)}{2\phi}=-\frac{y\theta(\mu)|_{\mu=y}-b(\theta(\mu))|_{\mu=y}-y\theta(\mu)+b(\theta(\mu))}{\phi}$$.

We need to calculate $$f(\hat y)\sqrt{2\pi\frac{1}{|h''(\hat y)|}}$$, where $$\hat y$$ is the mode (stationary point) of $$h(y)$$. (This is the Saddle-Point/Laplace approx. for the bounding constant).

$$h'(y) = -\frac{1}{\phi}(\theta(\mu)|_{\mu=y} + y \frac{d\theta}{d\mu}|_{\mu=y}- \frac{db}{d\theta} \frac{d\theta}{d\mu}|_{\mu=y}-\theta(\mu))$$

Now $$\frac{d\theta}{d\mu} = \frac{1}{V(\mu)}$$, and $$\frac{db}{d\theta}=\mu$$, evaluated at $$\mu=y$$ we get that the two middle terms cancel out. And we get that for the mode, $$\theta(\mu)|_{\mu=\hat y} = \theta(\mu)$$. This means that $$h(\hat y) = 0$$, and $$f(\hat y)$$ = 1.

Taking the 2nd derivative we get:

$$h''(y) = -\frac{1}{\phi}\frac{d\theta}{d\mu}|_{\mu=y} = -\frac{1}{\phi V(y)}$$

Hence the approximation to the f function integral comes out to be: $$f(\hat y)\sqrt{2\pi\frac{1}{|h''(\hat y)|}} = \sqrt{2\pi \phi V(y)}$$, and so the b-function is approximated by $$\frac{1}{\sqrt{2\pi \phi V(y)}}$$