The spatial arrangement of cones in the fovea: Bayesian analysis

Is the distribution of cones in the fovea random? We can never prove that this is so; we can only assess how probable this theory is relative to explicit alternative models, given the data. That paper does so for one simple alternative model of correlation. Readers of Mollon and Bowmaker’s letter in Nature may have felt sceptical about their assertion that the distribution of long and middle wave cones ‘is random’ given their data. Their χ test only used about a third of the information in the data about neighbouring cones. It would seem desirable to make full use of the data. Furthermore it might be argued that it is never possible to show that a phenomenon is random — only that the alternative models that have been studied for the phenomenon are less probable than the random model. In a Bayesian approach, explicit alternative models are constructed, and their relative probabilities can be evaluated in the light of the data. The Bayesian approach makes full use of all the relevant information in the data, and can be applied to any data set, no matter how quirky. The method of inference is mechanical once the alternative models have been defined. Bayesian inference automatically penalises excess parameters, so there is no need to fear being ‘duped’ into accepting over-complex models. Three models are studied to account for the data in [1]. All three models ignore ‘short’ (blue) cones, treating them as vacancies in a lattice of ‘long’ and ‘medium’ cones (henceforward referred to colloquially as red and green cones). H1: random The null hypothesis H1 is that the red and green cones occur independently with fixed probability. Thus the probability of a particular arrangement of cells, x, is: