Derivative Approximation for LIkelihood (DALI)

This approximation is based on Taylor expansion of the logarithm of the likelihood but being ordered in terms of order of derivatives:

\[\begin{split}log\mathcal{L} \approx log\mathcal{L}_{0}& -\left[\frac{1}{2}\sum_{i,j}\left\langle \partial_{i}h(\theta)|\partial_{j}h(\theta)\right\rangle _{0}\Delta\theta_{i}\Delta\theta_{j}\right] \\ & -\left[\frac{1}{2}\sum_{i,j,k}\left\langle \partial_{i}h(\theta)|\partial_{j}\partial_{k}h(\theta)\right\rangle _{0}\Delta\theta^{ijk} +\frac{1}{8}\sum_{i,j,k,l}\left\langle \partial_{i}\partial_{j}h(\theta)|\partial_{k}\partial_{l}h(\theta)\right\rangle _{0}\Delta\theta^{ijkl}\right] \\ & -\left[\frac{1}{6}\sum_{i,\cdots, l}\left\langle \partial_{i}h|\partial_{j}\partial_{k}\partial_{l}h\right\rangle \Delta\theta^{ijkl} +\frac{1}{12}\sum_{i,\cdots, m}\left\langle \partial_{i}\partial_{j}h|\partial_{k}\partial_{l}\partial_{m}h\right\rangle \Delta\theta^{ijklm} \right.\\ & \left. +\frac{1}{72}\sum_{i,\cdots, n}\left\langle \partial_{i}\partial_{j}\partial_{k}h|\partial_{l}\partial_{m}\partial_{n}h\right\rangle \Delta\theta^{ijklmn} \right] \\ & +\mathcal{O}(\partial^{4})\end{split}\]

where,

\[\begin{split}\Delta\theta^i =& \theta^i-\theta_0^i \\ \Delta\theta^{ij} =& \Delta\theta^i\cdot\Delta\theta^j \\ \Delta\theta^{ij..m} =& \Delta\theta^{i}\cdot\Delta\theta^j\cdots\Delta\theta^m\end{split}\]

GWDALI deals with derivatives through numerical derivatives with finite differences. It is availble to ways on computing these derivatives:

• With an uncertainty of \(O(h^2)\):

  \[\frac{df}{dx}=\left\{ \frac{1}{h}\left[f(x+h/2)-f(x-h/2)\right]\right\} +\mathcal{O}(h^{2})\]

• With uncertainty of \(O(h^4)\):

  \[\frac{df}{dx}=\left\{ \frac{4}{3h}\left[f(x+h/2)-f(x-h/2)\right]-\frac{1}{6h}\left[f(x+h)-f(x-h)\right]\right\} +\mathcal{O}(h^{4})\]

where \(h\) is the integration step.