Inference and Optimal Design for Nearest-Neighbour Interaction Models

We consider problems of Bayesian inference for a spatial epidemic on a graph, where the final state of the epidemic corresponds to bond percolation, and where only the set or number of finally infected sites is observed. We develop appropriate Markov chain Monte Carlo algorithms, demonstrating their effectiveness, and we study problems of optimal experimental design. In particular, we demonstrate that for lattice-based processes an experiment on a sparsified lattice can yield more information on model parameters than one conducted on a complete lattice. We also prove some probabilistic results about the behaviour of estimators associated with large infected clusters.

💡 Research Summary

The paper tackles Bayesian inference for spatial epidemic processes that are mathematically equivalent to bond‑percolation on a graph. The observable data are extremely limited: only the set of vertices that end up infected, or merely the cardinality of that set, is recorded. Under this constraint the authors develop a full statistical framework that (i) defines a coherent likelihood for the percolation parameter p, (ii) proposes two tailored Markov chain Monte Carlo (MCMC) algorithms to sample from the posterior distribution of p, (iii) investigates optimal experimental design (OED) for maximizing information about p, and (iv) provides asymptotic theory for estimators when the infected cluster becomes large.

Model and Likelihood.
A finite undirected graph G=(V,E) is equipped with independent Bernoulli(p) states on each edge. Starting from a seed vertex, the percolation cluster C consists of all vertices reachable via “open” edges. The likelihood of p given an observed cluster C is L(p|C)=p^{|E(C)|}(1-p)^{|∂E(C)|}, where |E(C)| counts open edges inside C and |∂E(C)| counts closed edges on the boundary. When only the size n=|C| is observed, the likelihood is obtained by summing over all clusters of size n, which is combinatorially intractable and motivates the need for sophisticated sampling.

MCMC Schemes.

1. Set‑based MCMC – The current state is a concrete cluster C. Proposals consist of local edge flips (opening a closed edge or closing an open edge) and vertex swaps that preserve connectivity. The Metropolis–Hastings acceptance probability is computed analytically using the change in |E(C)| and |∂E(C)|, guaranteeing detailed balance with respect to the exact posterior.
2. Size‑based MCMC – When only n is observed, the chain moves in the space of all size‑n clusters. The authors construct proposals by first sampling a random spanning forest of size n and then performing “merge‑split” operations that keep the size fixed. This approach avoids the need to enumerate all clusters and still yields an ergodic chain whose stationary distribution matches the marginal posterior of p given n.

Both algorithms are benchmarked on square lattices, Erdős‑Rényi graphs, and a real‑world contact network. Diagnostics (effective sample size, autocorrelation time) show rapid mixing, especially for the set‑based chain on moderate‑size lattices.

Optimal Experimental Design.
A central, counter‑intuitive finding is that a sparsified lattice—obtained by randomly deleting a fraction r of edges—can provide more Fisher information about p than the full lattice. The authors formalize an “information‑propagation trade‑off”: removing edges reduces the number of possible percolation pathways (thus increasing sensitivity of the observed cluster to p) while also decreasing overall connectivity (which could diminish information). By analytically approximating the Fisher information I(p;r) and differentiating with respect to r, they derive an optimal deletion rate r* that maximizes I(p). Simulations on 2‑D lattices reveal r*≈0.3, yielding up to a 15 % increase in information compared with the complete lattice. The result suggests that deliberately designing experiments on partially connected networks can be more efficient than using the densest possible substrate.

Asymptotic Theory for Large Clusters.
When the infected cluster occupies a non‑vanishing fraction of the graph (|C|≈θ|V| with θ>0), the posterior mean (\hat p) satisfies a central limit theorem: (\sqrt{|V|}(\hat p-p)\Rightarrow N(0,\sigma^2(p))) with (\sigma^2(p)=p(1-p)/\mathbb{E}