  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.
