In recent years there has been considerable activity in the development and application of Bayesian inferential methods for infectious disease data using stochastic epidemic models. Most of this activity has employed computationally intensive approaches such as Markov chain Monte Carlo methods. In contrast, here we address fundamental questions for Bayesian inference in the setting of the standard SIR (Susceptible-Infective-Removed) epidemic model via simple methods. Our main focus is on the basic reproduction number, a quantity of central importance in mathematical epidemic theory, whose value essentially dictates whether or not a large epidemic outbreak can occur. We specifically consider two SIR models routinely employed in the literature, namely the model with exponentially distributed infectious periods, and the model with fixed length infectious periods. It is assumed that an epidemic outbreak is observed through time. Given complete observation of the epidemic, we derive explicit expressions for the posterior densities of the model parameters and the basic reproduction number. For partial observation of the epidemic, when the entire infection process is unobserved, we derive conservative bounds for quantities such as the mean of the basic reproduction number and the probability that a major epidemic outbreak will occur. If the time at which the epidemic started is observed, then linear programming methods can be used to derive suitable bounds for the mean of the basic reproduction number and similar quantities. Numerical examples are used to illustrate the practical consequences of our findings. In addition, we also examine the implications of commonly-used prior distributions on the basic model parameters as regards inference for the basic reproduction number.