Abstract

Inference and optimal design problems for finite clusters from percolation on the integer lattice Z^d or, equivalently, for SIR epidemics evolving on a bounded subset of Z^d with constant life times are considered. The corresponding percolation probability p is considered to be unknown, possibly depending, through the experimental design, on other parameters. We consider inference under each of the following two scenarios:

(i) The observations consist of the set of sites which are ever infected, so that the routes by which infections travel are not observed (in terms of the bond percolation process, this corresponds to a knowledge of the connected component containing the initially infected site—-the location of this site within the component not being relevant to inference for p).

(ii) All that is observed is the size of the set of sites which are ever infected.

We discuss practical aspects of Bayesian utility-based optimal designs for the former scenario and prove that the sequence of MLE ’s for p converges to the critical percolation probability p_c under the latter scenario (when the size of the finite cluster grows).

This is a joint work with Professor Gavin Gibson and Dr Stan Zachary, both with Heriot-Watt University, Edinburgh.