Midpoint rule approximation

The probability of the model is the integral over $h(\theta)$. It can be approximated to $h(\theta)s(\theta)$ using the midpoint rule from numerical integration. However, this approximation assumes that $h(\theta)$ is relatively ``flat'' about $\theta$. If this is not the case, then it might be possible for $h(\theta)s(\theta)$ to become greater than 1. This is clearly absurd since $h(\theta)s(\theta)$, represents a probability. This is a clear indication that this MML approximation is no longer justified. However, Wallace [#!Wallace:2000!#] has suggested replacing $h(\theta)s(\theta)$ by a new term $q(\theta)$.

$$q(\theta) = \frac{1}{\sqrt{1+[\frac{1}{h(\theta)s(\theta)}]^2}}$$ (8)

There are simple and obvious extensions to the above expression, such as replacing $q(\theta)$ by 1-e^-x, or generalising it to $\Big[ 1+[\frac{1}{h(\theta)s(\theta)}]^n \Big]^{-1/n}$ for large n's. These alterations will ensure that $q(\theta)$ never exceeds one. However, when $h(\theta)s(\theta)$ is close to zero, it has little effect.