Taylor Series approximation

For the second part of the message, the approximation for the first part of the message (subsection ) could also be applied. This will result in the following.

$$\frac{1}{s(\theta)} [ -\log_e f(x\vert\theta) ] \times s(\theta) = -\log_e f(x\vert\theta)$$ (9)

Indeed, this is the first order approximation of the technique that Wallace and Freeman [#!Wallace.Freeman:1987!#] actually used. However, this does not achieve sufficient accuracy and with a greater deal of work, a more useful second-order approximation can be derived.

Instead, the term $-\log_e f(x\vert\theta^\prime)$ is expanded using Taylor series about the point $\theta$. Since $\theta \approx \theta^\prime$, the higher order terms will approach zero.

$$\begin{split} -\log_e f(x\vert\theta^\prime) &= -\log_e f(x\... ...\partial \theta^n } (-\log_e f(x\vert\theta)) + ... \end{split}$$ (10)

Wallace and Freeman [#!Wallace.Freeman:1987!#] merely took the approximation to the second order. Later in this thesis, the fourth order extension will be derived (section ). However, when the above expression is truncated at the second order term and simplified, the message length of the second part is retrieved.

$$\frac{1}{s(\theta)} \int_ {\theta - \frac{s(\theta)}{2}}^{\t... ...- \log_e f(x\vert\theta) + \frac{s(\theta)^2}{24} F(x, \theta)$$ (11)

where $F(x, \theta) = \frac{\partial^2}{\partial \theta^2} -\log_e f(x\vert\theta)$

At this stage, the total message length is

$$\textrm{MesgLen~} = -\log_e h(\theta)s(\theta) - \log_e f(x\vert\theta) + \frac{s(\theta)^2}{24} F(x, \theta)$$ (12)