# Maximum Entropy and the Black Object

## Neil Pendock, Peter Fridjhon and Michael Sears

Department of Computational and Applied Mathematics

University of the Witwatersrand

Johannesburg, South Africa

### Abstract

We consider the problem of estimating the number of rhinos in a game
reserve from incomplete census data. We start by dividing the game
reserve into **k** cells and argue that rhinos may be
sighted in a cell independently and at random implying a Poisson
observation distribution. **{n_i}** is the distribution
of rhinos in the cells. This simple formulation assumes that the
expected number of rhinos is directly proportional to the size of the
cells. The **{n_i}** will more generally depend on the
diverse nature of the game reserve (for example vegetation cover,
rhino behavior etc.) and thus have a prior distribution
**{m_i}** which is not necessarily the uniform
distribution. We derive an entropic prior distribution for the
**{n_i}** in terms of **{m_i}**.
The set of occupation numbers for rhinos in the cells
**{n_i}** constitutes a nearly black image as most of the
cells will be empty. Some of the cells are censused using aerial
surveys and random walks through the bush. The data
**{d_i}** are incomplete and noisy. We assume a Poisson
likelihood distribution for **p({d_i}|{n_i})** and use
Bayes' theorem to express the posterior distribution for
**p({n_i}|{d_i})** in terms of the entropic prior and
likelihood.

To estimate the marginal distribution **p({n_i}|{d_i})**
we construct a Markov chain with limit distribution
**p({n_i}|{d_i})**. We visit each of the
**k** cells in turn and replace the current occupation
value **n_i** by a new value sampled from the conditional
distribution **p(n_i|{n_j}\n_i,{d_j})**. In addition, the
overall evidence **p({d_i})** may be calculated.

We apply the above methodology to a data set collected at the Umfolozi
game reserve in Kwazulu/Natal, South Africa, over several years.

MaxEnt 94 Abstracts / mas@mrao.cam.ac.uk