Inferring diffusivity from killed diffusion
This work addresses a specific inference problem in statistical physics or biophysics, with incremental contributions to Bayesian methods for high-dimensional Poisson data.
The authors tackled the problem of inferring an unknown diffusivity parameter from observed positions where diffusing molecules bind and stop moving, showing that the inference can be consistently solved using Bayesian methods under certain conditions on the binding potential. They provided a numerical illustration using MCMC methods.
We consider diffusion of independent molecules in an insulated Euclidean domain with unknown diffusivity parameter. At a random time and position, the molecules may bind and stop diffusing in dependence of a given `binding potential'. The binding process can be modeled by an additive random functional corresponding to the canonical construction of a `killed' diffusion Markov process. We study the problem of conducting inference on the infinite-dimensional diffusion parameter from a histogram plot of the `killing' positions of the process. We show first that these positions follow a Poisson point process whose intensity measure is determined by the solution of a certain Schrödinger equation. The inference problem can then be re-cast as a non-linear inverse problem for this PDE, which we show to be consistently solvable in a Bayesian way under natural conditions on the initial state of the diffusion, provided the binding potential is not too `aggressive'. In the course of our proofs we obtain novel posterior contraction rate results for high-dimensional Poisson count data that are of independent interest. A numerical illustration of the algorithm by standard MCMC methods is also provided.