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This paper presents Bayesian parameter estimation for first order Grey systemmodels' parameters (or sometimes referred to as hyperparameters). There aredifferent 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 $GM(1,1| \cos(\omegat), \sin(\omega t)$. The whitenization equation of these models is afirst-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 $GM(1,1| cos(\omega t)$, $f(t) = b_1\sin(\omegat)+b_2$ in $GM(1,1| \sin(\omega t)$, $f(t) = b_1\sin(\omega t) + b_2\cos(\omegat) + b_3$ in $GM(1,1| \cos(\omega t), \sin(\omega t)$, $f(t) = b x^2$ in GreyVerhulst model (GVM),  and where $b, b_1, b_2$, and $b_3$ are parameters. The results from Bayesianestimations are compared to the least square estimated models with fixed$\omega$. We found that using rolling Bayesian estimations for GM parameterscan allow us to estimate the parameters in all possible forms. Based on thedata used for the comparison, the numerical results showed that models withBayesian parameter estimations are up to 45\% more accurate in mean squarederrors.

Bayesian Parameter Estimations for Grey System Models in Online Traffic Speed Predictions