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Bayes estimation methods are proposed in this paper to estimate the unknown parameters of the Alpha Power Inverted Exponential (APIE) distribution for complete sample and this distribution was introduced by Unal, Cakmakyapan and Ozel (2018). Lindley's approximation, Tierney and Kadane's (T-K) approximation and Markov Chain Monte Carlo (MCMC) method are used for obtaining bayes estimators with squared error loss function (SELF) and Gamma prior for the unknown parameters of this distribution. The performance of these estimators are compared with maximum likelihood estimators by simulation study and the comparison is based on MSE's. Also two real life data sets are used to illustrate the usefulness of the Alpha Power Inverted Exponential (APIE) distribution. Different criterions are used to access the goodness of fit of APIE distribution as compared to other distributions. [ABSTRACT FROM AUTHOR]