  Uncertainties from experiments and models render multi-modal difficulties in
model calibrations. Bayesian inference and \textsc{mcmc} algorithm have been
applied to obtain posterior distributions of model parameters upon uncertainty.
However, multi-modality leads to difficulty in convergence criterion of
parallel \textsc{mcmc} sampling chains. The commonly applied $\widehat{R}$
diagnostic does not behave well when multiple sampling chains are evolving to
different modes. Both partitional and hierarchical clustering methods has been
combined to the traditional $\widehat{R}$ diagnostic to deal with sampling of
target distributions that are rough and multi-modal. It is observed that the
distributions of binding parameters and pore diffusion of particle parameters
are multi-modal. Therefore, the steric mass-action model used to describe
ion-exchange effects of the model protein, lysozyme, on the \textsc{sp}
Sepharose \textsc{ff} stationary phase might not be fully capable in certain
experimental conditions, as model uncertainty from steric mass-action would
result in multi-modality.
