Binomial Distribution

When MML is applied to a particular problem, it is common to categorise the problem by identifying the parameters involved and the resulting probability distribution - whether they be discrete or continuous. Generally speaking, the problem is specified by the prior function - $h(\theta)$ and the likelihood function - $f(x\vert\theta)$ or $f(m\vert\theta)$.

The binomial problem describes the generation of data with two distinct states, (eg 0-state/1-state or black/white etc). The first state (eg 0-state or black) occurs m times with probability $\theta$ and the second state occurs ntimes with probability $(1-\theta)$. Therefore, there are N=m+n data-points.

$$\begin{split} h(\theta) &= 1 \\ f(m\vert\theta) &= {N \choose m} \theta^m (1-\theta)^n \end{split}$$ (52)

Given the above information, $F(\theta)$ and $G(\theta)$ can be shown to be

$$\begin{split} F(\theta) &= \frac{N}{\theta(1-\theta)} \\ G(... ...a) &= \frac{6N}{\theta^3} + \frac{6N}{(1-\theta)^3} \end{split}$$ (53)