  Assessing the quality of parameter estimates for models describing the motion
of single molecules in cellular environments is an important problem in
fluorescence microscopy. We consider the fundamental data model, where
molecules emit photons at random times and the photons arrive at random
locations on the detector according to complex point spread functions (PSFs).
The random, non-Gaussian PSF of the detection process and random trajectory of
the molecule make inference challenging. Moreover, the presence of other nearby
molecules causes further uncertainty in the origin of the measurements, which
impacts the statistical precision of estimates. We quantify the limits of
accuracy of model parameter estimates and separation distance between closely
spaced molecules (known as the resolution problem) by computing the Cramer-Rao
lower bound (CRLB), or equivalently the inverse of the Fisher information
matrix (FIM), for the variance of estimates. This fundamental CRLB is crucial,
as it provides a lower bound for more practical scenarios. While analytic
expressions for the FIM can be derived for static molecules, the analytical
tools to evaluate it for molecules whose trajectories follow SDEs are still
mostly missing. We address this by presenting a general SMC based methodology
for both parameter inference and computing the desired accuracy limits for
non-static molecules and a non-Gaussian fundamental detection model. For the
first time, we are able to estimate the FIM for stochastically moving molecules
observed through the Airy and Born & Wolf PSF. This is achieved by estimating
the score and observed information matrix via SMC. We sum up the outcome of our
numerical work by summarising the qualitative behaviours for the accuracy
limits as functions of e.g. collected photon count, molecule diffusion, etc. We
also verify that we can recover known results from the static molecule case.
