Profile likelihood analysis for a stochastic model of diffusion in heterogeneous media

TL;DR

This paper develops methods to estimate layer-specific diffusion rates in layered media using profile likelihoods derived from stochastic models, aiding in parameter identification from heat conduction data.

Contribution

It introduces both exact and approximate likelihood approaches for parameter inference in a layered stochastic diffusion model, including strategies for unidentifiable parameters.

Findings

Exact likelihood via Markov chain is computationally intensive.

Gamma distribution approximation effectively estimates exit time distributions.

Reduced models help identify parameters when original model is unidentifiable.

Abstract

We compute profile likelihoods for a stochastic model of diffusive transport motivated by experimental observations of heat conduction in layered skin tissues. This process is modelled as a random walk in a layered one-dimensional material, where each layer has a distinct particle hopping rate. Particles are released at some location, and the duration of time taken for each particle to reach an absorbing boundary is recorded. To explore whether this data can be used to identify the hopping rates in each layer, we compute various profile likelihoods using two methods: first, an exact likelihood is evaluated using a relatively expensive Markov chain approach; and, second we form an approximate likelihood by assuming the distribution of exit times is given by a Gamma distribution whose first two moments match the expected moments from the continuum limit description of the stochastic model. Using the exact and approximate likelihoods we construct various profile likelihoods for a range of problems. In cases where parameter values are not identifiable, we make progress by re-interpreting those data with a reduced model with a smaller number of layers.