Epidemics of seasonal influenza inflict a huge burden in temperate climes such as Melbourne (Australia) where there is also significant variability in their timing and magnitude. Particle filters combined with mechanistic transmission models for the spread of influenza have emerged as a popular method for forecasting the progression of these epidemics. Despite extensive research it is still unclear what the optimal models are for forecasting influenza, and how one even measures forecast performance. In this paper, we present a likelihood-based method, akin to Bayes factors, for model selection when the aim is to select for predictive skill. Here, "predictive skill" is measured by the probability of the data after the forecasting date, conditional on the data from before the forecasting date. Using this method we choose an optimal model of influenza transmission to forecast the number of laboratory-confirmed cases of influenza in Melbourne in each of the 2010-15 epidemics. The basic transmission model considered has the susceptible-exposed-infectious-recovered structure with extensions allowing for the effects of absolute humidity and inhomogeneous mixing in the population. While neither of the extensions provides a significant improvement in fit to the data they do differ in terms of their predictive skill. Both measurements of absolute humidity and a sinusoidal approximation of those measurements are observed to increase the predictive skill of the forecasts, while allowing for inhomogeneous mixing reduces the skill. We discuss how our work could be integrated into a forecasting system and how the model selection method could be used to evaluate forecasts when comparing to multiple surveillance systems providing disparate views of influenza activity.