  This paper presents Bayesian parameter estimation for first order Grey system
models' parameters (or sometimes referred to as hyperparameters). There are
different forms of first-order Grey System Models. These include $GM(1,1)$,
$GM(1,1| \cos(\omega t)$, $GM(1,1| \sin(\omega t)$, and $GM(1,1| \cos(\omega
t), \sin(\omega t)$. The whitenization equation of these models is a
first-order linear differential equation of the form \[ \frac{dx}{dt} + a x =
f(t) \] where $a$ is a parameter and $f(t) = b$ in $GM(1,1|)$ , $f(t) =
b_1\cos(\omega t) + b_2$ in $GM(1,1| cos(\omega t)$, $f(t) = b_1\sin(\omega
t)+b_2$ in $GM(1,1| \sin(\omega t)$, $f(t) = b_1\sin(\omega t) + b_2\cos(\omega
t) + b_3$ in $GM(1,1| \cos(\omega t), \sin(\omega t)$, $f(t) = b x^2$ in Grey
Verhulst model (GVM),
  and where $b, b_1, b_2$, and $b_3$ are parameters. The results from Bayesian
estimations are compared to the least square estimated models with fixed
$\omega$. We found that using rolling Bayesian estimations for GM parameters
can allow us to estimate the parameters in all possible forms. Based on the
data used for the comparison, the numerical results showed that models with
Bayesian parameter estimations are up to 45\% more accurate in mean squared
errors.
