Poisson Distribution

In additional to the binomial case studied within the thesis, the Poisson distribution was also briefly investigated. However, the produces of these investigations are either incomplete or are re-derivations of previously published material.

The Poisson distribution has a prior and likelihood function of

$$\begin{split} h(r) &= \frac{1}{\alpha}e^{-r/\alpha} \\ f(\ve... ...\prod_{i=1}^N e^{-r t_i} \frac{(r t_i)^{c_i}}{c_i!} \end{split}$$ (66)

For $c_i = [0, 1, 2, 3, ...)~\forall~i$ and r and t_i are positive, real valued variables

The data we are trying to encode is in a list of 2-tuples as $(c_1, t_1); (c_2, t_2); \ldots; (c_N, t_N)$. That is, the process we are trying to model was sampled at times $0, t_1, t_1+t_2, ..., \sum_{i=1}^N t_i$and we have observed a count of c₁ between times t₁ and 0, c₂ between times t₁+t₂ and t₁ and so on.

Therefore, the observed Fisher information can be determined as -

$$\begin{split} \frac{\partial^2}{\partial r^2} [ -\log_e f(\v... ...\\ &= \sum_{i=1}^N \frac{c_i}{r^2} = \frac{C}{r^2} \end{split}$$ (67)

where $C = \sum_{i=1}^N c_i$ and $T = \sum_{i=1}^N t_i$. When we take its expectation value, we get the expected Fisher information, $\frac{T}{r}$.

This means the message length is

$$\begin{split} \textrm{MesgLen~} &= -\log_e \frac{h(r) f(\vec... ...{1}{2}\log_e r - \log_e \sqrt{12} + \frac{1}{2} \\ \end{split}$$ (68)

Therefore, $\hat{r}$ can be calculated by minimising the message length with respect to r.

$$\begin{split} 0 &= \frac{1}{\alpha} + \sum_{i=1}^N \Big[ t_i... ...\hat{r} &= \frac{C+\frac{1}{2}}{T+\frac{1}{\alpha}} \end{split}$$ (69)

The fourth order can be derived by finding $\frac{h_r(r)}{h(r)}$, $\frac{f_r(\vec{c}\vert r)}{f(\vec{c}\vert r)}$, F(r), F_r(r), G(r), G_r(r).

$$\begin{split} \frac{h_r(r)}{h(r)} &= \frac{-1}{\alpha} \\ \f... ...(r)^2 G_r(r)}{G(r)^2 \sqrt{F(r)^2+\frac{6}{5}G(r)}} \end{split}$$ (70)

These expressions can be substituted into equation () with the expressions for $z(r) = z(\theta)$ and $z_r(r) = z_\theta(\theta)$ at equation () and () respectively.

To apply MMLD to the Poisson distribution, the algorithm in section can be applied unaltered. The expressions for h(r) and $f(\vec{c}\vert r)$ are above. The expression for the maximum likelihood is simply $\frac{C}{T}$

For point estimation, an expression for p(y|r) is the probability of future data y given the rate r. This is simply the expression $e^{-r t_i} \frac{(r t_i)^{y}}{y!}$ The variable t_i is simply the length of time until the next time interval - this value will be discussed later.