  The abnormal aggregation of extracellular amyloid-$\beta$ (A\beta) in senile
plaques resulting in calcium (Ca^{+2}) dyshomeostasis is one of the primary
symptoms of Alzheimer's disease (AD). Significant research efforts have been
devoted in the past to better understand the underlying molecular mechanisms
driving A\beta deposition and Ca^{+2} dysregulation. To better understand this
interaction, we report a novel stochastic model where we analyze the positive
feedback loop between A\beta and Ca^{+2} using ADNI data. A good therapeutic
treatment plan for AD requires precise predictions. Stochastic models offer an
appropriate framework for modelling AD since AD studies are observational in
nature and involve regular patient visits. The etiology of AD may be described
as a multi-state disease process using the approximate Bayesian computation
method. So, utilizing ADNI data from $2$-year visits for AD patients, we employ
this method to investigate the interplay between A\beta and Ca^{+2} levels at
various disease development phases. Incorporating the ADNI data in our
physics-based Bayesian model, we discovered that a sufficiently large
disruption in either A\beta metabolism or intracellular Ca^{+2} homeostasis
causes the relative growth rate in both Ca^{+2} and A\beta, which corresponds
to the development of AD. The imbalance of Ca^{+2} ions causes A\beta disorders
by directly or indirectly affecting a variety of cellular and subcellular
processes, and the altered homeostasis may worsen the abnormalities of Ca^{+2}
ion transportation and deposition. This suggests that altering the Ca^{+2}
balance or the balance between A\beta and Ca^{+2} by chelating them may be able
to reduce disorders associated with AD and open up new research possibilities
for AD therapy.
