Malmquist productivity index numbers have been extensively used in productivity analysis. These indices have an advantage over traditional Fisher and Tornqvist productivity indices because they allow productivity to be decomposed into technical change and technical efficiency change using only quantity information. Common methods for estimating the component distance functions of this index include data envelopment analysis (DEA) and stochastic frontier analysis (SFA). The non-statistical nature of DEA does not allow appropriate measures of reliability to be easily computed for the productivity index. Moreover, SFA in a sampling theory framework yields estimators with properties that are unknown in small samples. This drawback becomes extremely acute when quantities of interest, such as Malmquist Index numbers, are nonlinear functions of structural parameters. The implementation of SFA in a Bayesian framework can be used to conveniently address these issues. In particular, the use of Bayesian SFA allows us to: i) obtain exact small sample results for the parameters of the model. ii) derive the full posterior distribution of any quantity of interest, including any function of individual efficiencies, parameters and/or the data. Based on these advantages, this dissertation uses Bayesian SFA to estimate the Malmquist productivity index and its components, and to compute the measures of reliability for the index. Implementation of the Bayesian approach requires the use of Markov chain Monte Carlo simulation methods. The productivity measurement techniques discussed in this study are illustrated using data on the electricity supply industry of Australia.