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A fundamental problem in analysis of complex systems is getting a reliable estimate of entropy of their probability distributions over the state space. This is difficult because unsampled states can contribute substantially to the entropy, while they do not contribute to the Maximum Likelihood estimator of entropy, which replaces probabilities by the observed frequencies. Bayesian estimators overcome this obstacle by introducing a model of the low-probability tail of the probability distribution. Which statistical features of the observed data determine the model of the tail, and hence the output of such estimators, remains unclear. Here we show that well-known entropy estimators for probability distributions on discrete state spaces model the structure of the low probability tail based largely on few statistics of the data: the sample size, the Maximum Likelihood estimate, the number of coincidences among the samples, the dispersion of the coincidences. We derive approximate analytical entropy estimators for undersampled distributions based on these statistics, and we use the results to propose an intuitive understanding of how the Bayesian entropy estimators work. 

The problem of deciphering how low-level patterns (action potentials in the brain, amino acids in a protein, etc.) drive high-level biological features (sensorimotor behavior, enzymatic function) represents the central challenge of quantitative biology. The lack of general methods for doing so from the size of datasets that can be collected experimentally severely limits our understanding of the biological world. For example, in neuroscience, some sensory and motor codes have been shown to consist of precisely timed multi-spike patterns. However, the combinatorial complexity of such pattern codes have precluded development of methods for their comprehensive analysis. Thus, just as it is hard to predict a protein’s function based on its sequence, we still do not understand how to accurately predict an organism’s behavior based on neural activity. Here, we introduce the unsupervised Bayesian Ising Approximation (uBIA) for solving this class of problems. We demonstrate its utility in an application to neural data, detecting precisely timed spike patterns that code for specific motor behaviors in a songbird vocal system. In data recorded during singing from neurons in a vocal control region, our method detects such codewords with an arbitrary number of spikes, does so from small data sets, and accounts for dependencies in occurrences of codewords. Detecting such comprehensive motor control dictionaries can improve our understanding of skilled motor control and the neural bases of sensorimotor learning in animals. To further illustrate the utility of uBIA, we used it to identify the distinct sets of activity patterns that encode vocal motor exploration versus typical song production. Crucially, our method can be used not only for analysis of neural systems, but also for understanding the structure of correlations in other biological and nonbiological datasets. 

Although different animal species often exhibit extensive variation in many behaviors, typically scientists examine one or a small number of behaviors in any single study. Here, we propose a new framework to simultaneously study the evolution of many behaviors. We measured the behavioral repertoire of individuals from six species of fruit flies using unsupervised techniques and identified all stereotyped movements exhibited by each species. We then fit a Generalized Linear Mixed Model to estimate the intra- and inter-species behavioral covariances, and, by using the known phylogenetic relationships among species, we estimated the (unobserved) behaviors exhibited by ancestral species. We found that much of intra-specific behavioral variation has a similar covariance structure to previously described long-time scale variation in an individual’s behavior, suggesting that much of the measured variation between individuals of a single species in our assay reflects differences in the status of neural networks, rather than genetic or developmental differences between individuals. We then propose a method to identify groups of behaviors that appear to have evolved in a correlated manner, illustrating how sets of behaviors, rather than individual behaviors, likely evolved. Our approach provides a new framework for identifying co-evolving behaviors and may provide new opportunities to study the mechanistic basis of behavioral evolution.