How biased are maximum entropy models of neural population activity?

Maximum entropy models have become popular statistical models in neuroscience and other areas of biology, and can be useful for quantifying the coding properties of sensory systems. However, maximum entropy models fit to small data sets can be subject to sampling bias; i.e. the true entropy of the system can be severely under-estimated. Here we study the sampling properties of estimates of the entropy obtained from maximum entropy models. We focus on the pairwise binary model, which is used extensively to model neural population activity. We show that if the data is well described by a pairwise model, the bias is equal to the number of parameters divided by twice the number of observations. However, if the higher order correlations in the data deviate from those predicted by the model, the bias can be larger. Using a phenomenological model of neural population recordings, we find that the additional bias due to higher-order correlations is largest for small firing probabilities, high correlations, and large population sizes. However, numerical studies indicate that even in the worst case, it is only about a factor of four larger. We derive guidelines for how much recording time one needs to achieve a bias which is smaller than some specified level of accuracy. Finally, we show how a modified plug-in estimate of the entropy can be used for bias correction