Schematic of the full three-dimensional model space, with dimensions indicating population mean (σ), population standard deviation (σ) and stimulus fundamental frequency (f). To ease computational demands, the model space was discretised. Each point in model space corresponds to the probability of a particular f, given a specific μ and σ, that is, P(f|μ,σ). Therefore each column (along the f dimension) gives the probability distribution P(f|μ,σ), and corresponds to the forward model. The planes of the model space, conversely, indicate the probability of each given combination of μ and σ, given a particular f value, that is, P(μ,σ|f), or in other words the inverse model. To generate priors based on a series of observed f values, the scalar product of the planes for each of these f values is taken, and the resulting plane scaled to a sum of 1. This plane represents the estimates of the hidden states μ and σ and is then used to weight the columns of the model space. The weighted model space is averaged into a single column (forward model), and scaled to a sum of 1, thus providing optimal priors on the assumption that the f population does not change. The priors assuming a population change are derived from the same procedure, but with uniform weighting across the model space. The change and no change priors are then weighted according to the probability of a population change, and then summed. The only part of the process not illustrated here is the inference of population changes, determining how many preceding f values form part of the model inversion, which is explained in Equation 2.