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