Bayesian inference provides an elegant normative framework for understanding the characteristic biases and discrimination thresholds in visual speed perception (Stocker & Simoncelli 2006). However, the framework is difficult to validate due to its flexibility and the fact that suitable constraints on the prior beliefs and the likelihood functions have been missing. Here we use assumptions of efficient coding to develop a better constrained Bayesian observer model (Wei & Stocker 2015). In the new model, the stimulus distribution links and jointly constrains both the likelihood function and the prior belief. We fit the model to existing psychophysical speed discrimination data, representing discrimination measurements over a broad range of speeds and stimulus uncertainties. Cross- validation confirms that the model fits the data as well as parametric approximations (Weibull fits) of each psychometric curve. The extracted prior beliefs are closely following power-law functions with an exponent of ~ -1 and are much more consistent across subjects than when extracted with the previous model. Furthermore, the efficient coding assumption of the model also makes the specific prediction that for a variable that follows a power-law distribution, the neural encoding space should be logarithmic. We tested this prediction by analyzing the speed encoding characteristics of a large population of MT neurons (Nover et al. 2005). We show that the prior predicted by the encoding characteristics of the neurons (neural prior) very closely matches the slow-speed power-law prior extracted from the behavioral data (behavioral prior). Our results demonstrate that a Bayesian observer model constrained by efficient coding not only accurately accounts for the behavioral characteristics of visual speed perception, but also provides a normative account of the logarithm encoding of MT neurons (which gives rise to Weber’s law) as the efficient neural representation of a power-law distributed perceptual variable.