The Saddlepoint.

The $ P\times 1$ saddlepoint vector $ \lambda$ is the solution to the equation

$\displaystyle {\bf A}^\prime {\bf K}_u^1 = {\bf z},$

where $ {\bf K}_u^1$ is the diagonal matrix with diagonal element $ i$ equal to $ k_i/u_i$, written compactly (borrowing efficient MATLAB notation) as

$\displaystyle {\bf K}_u^1 = {\rm diag}({\bf k}./ {\bf u}),$

where $ u_i$ are the elements of vector $ {\bf u}$ defined by

$\displaystyle {\bf u}= {\bf 1}-2{\bf A}$$ \lambda$$\displaystyle ,$

where $ {\bf 1}$ is a vector of 1's.

The saddlepoint approximation for $ \log p({\bf z})$ is given by

$\displaystyle l_{sp}=\frac{
-P\log(2\pi) - \log \det(2{\bf G}) -{\bf k}^\prime \log({\bf u})
-2 \mbox{\boldmath $\lambda$}^\prime {\bf z}}{2},$

where $ \log$ operates separately on each element and

$\displaystyle {\bf G}={\bf A}^\prime {\bf K}_u^2 {\bf A},$

where

$\displaystyle {\bf K}_u^p = {\rm diag}({\bf k}./ {\bf u}^{\cdot p} ),$

where $ {\bf u}^{\cdot p}$ borrows the efficient MATLAB notation for element-wise exponentiation by $ p$.

To find the saddlepoint, we initialize $ \lambda$ to zeros, then implement the iterative Newton-Raphson search for the saddlepoint as described in [16]. For convenience, a MATLAB implementation is provided below in section 16.6.3.



Baggenstoss 2017-05-19