In any Bayesian inference problem, the posterior distribution is a product of the likelihood and the prior: thus, it is affected by both in cases where one possesses little or no information about the target parameters in advance. In the case of an objective Bayesian analysis, the resulting posterior should be expected to be universally agreed upon by everyone, whereas . subjective Bayesianism would argue that probability corresponds to the degree of personal belief. In this paper, we consider Bayesian estimation of two-parameter exponential distribution using the Bayes approach needs a prior distribution for parameters. However, it is difficult to use the joint prior distributions. Sometimes, by using linear transformation of reliability function of two-parameter exponential distribution in order to get simple linear regression model to estimation of parameters. Here, we propose to make Bayesian inferences for the parameters using non-informative priors, namely the (dependent and independent) Jeffreys’ prior and the reference prior. The Bayesian estimation was assessed using the Monte Carlo method. The criteria mean square error was determined evaluate the possible impact of prior specification on estimation. Finally, an application on a real dataset illustrated the developed procedures.

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