Estimating the orders of  bivariate mixed autoregressive moving average processes, denoted by ARMA₂ (p,q) ,  is the first and one of the most important phases in time series analysis. This article has three different objectives. The first one is to develop an indirect Bayesian methodology to estimate the orders of  bivariate mixed ARMA Processes. Assuming the maximum orders are known, the indirect methodology is based on approximating the posterior distribution of the model's coefficients by a matrix t distribution. Then one may test the significance of the  coefficients marginally or conditionally and eliminate insignificance. The second objective is to develop a pure  Bayesian  methodology to estimate the orders  ARMA₂ (p,q)  Processes. The pure  methodology is based on deriving an approximate joint posterior probability mass function of the orders in a convenient from. Then one may inspect the posterior probabilities and select the orders with maximum probability to be the estimated orders. The third objective is to carry out a simulation study to assess the performance and numerical efficiency of the two proposed methodologies and compare the results with the well-known automatic technique AIC or Akaike's information criterion. The numerical results show that the proposed indirect methodology is the best and can efficiently estimate the orders of bivariate mixed ARMA processes with high precision for moderate and large time series lengths.