  Systems driven by Brownian motion are ubiquitous. A prevailing challenge is
inferring, from data, the diffusion and kinetic parameters that describe these
stochastic processes. In this work, we investigate a multi-state diffusion
process that arises in the context of single particle tracking (SPT), wherein
the motion of a particle is governed by a discrete set of diffusive states, and
the tendency of the particle to switch between these states is modeled as a
random process. We consider two models for this behavior: a mixture model and a
hidden Markov model (HMM). For both, we adopt a Bayesian approach to sample the
distributions of the underlying parameters and implement a Markov Chain Monte
Carlo (MCMC) scheme to compute the posterior distributions, as in Das, Cairo,
Coombs (2009). The primary contribution of this work is a study of the
robustness of this method to infer parameters of a three-state HMM, and a
discussion of the challenges and degeneracies that arise from considering three
states. Finally, we investigate the problem of determining the number of
diffusive states using model selection criteria. We present results from
simulated data that demonstrate proof of concept, as well as apply our method
to experimentally measured single molecule diffusion trajectories of monomeric
phosphoinositide-dependent kinase-1 (PDK1) on a synthetic target membrane where
it can associate with its binding partner protein kinase C alpha isoform
(PKC$\alpha$) to form a heterodimer detected by its significantly lower
diffusivity.
  All matlab software is available here:
\url{https://github.com/MathBioCU/SingleMolecule}
