Bayesian analysis of the exponential model, based on life tests that are terminated at preassigned time points or after preassigned number of failures, has been developed. For the prior distribution of the parameter involved, uniform, inverted gamma and exponential densities have been examined. The estimation of the reliability function has also been carried out by using Bayesian methods and the case of ‘attribute testing’ has been considered briefly. The role of prior quasi-densities when a life tester has no prior information has been illustrated and it has been observed that the reliability estimate for a diffuse prior which is uniform over the entire positive real line closely resembles the classical MVU estimate obtained by Pugh. It has also been noted that the Bayes estimate of the exponential parameter θ for a prior quasi-density of the form 1/θ2coincides with the classical MVU estimate of Epstein and Sobel. Further, it has been proved that in a wide class of prior densities, proper or improper, 1/θ2is the only prior which leads to the MVU estimate.