Analyzing an organism's sensors using Maximum Entropy models with bias, variance, and confusion matrices

Biological organisms have sensors that communicate information about the environment, be it a noisy copy of the current environmental state, memory of a past environmental state, or predictive information about the environment's future. Analyzing how well these biological sensors function has usually been done with mutual information(s). We suggest that alternatively, by viewing the remainder of the organism as having a homunculus, one may profitably analyze these evolved biosensors using bias and variance or confusion matrices, depending on if the environment is ordered or categorical. Stimulus-dependent Maximum Entropy models (which are sometimes actually just statistical mechanical models) are used to develop estimators of the environmental state given the sensor state, and these in turn are then used to calculate the bias and variance of the estimator, or confusion matrices. We focus on several examples to understand the utility of non-information-based analyses: one classic genetic regulatory circuit from E. coli analyzed using statistical mechanics that leads to bias and variance calculations for exactly how much lac repressor exists, several instantiations of the ubiquitous Monod-Wyman-Changeux model, and neural activity from stimulus-dependent Maximum Entropy models that leads to confusion matrices for understanding what errors an organoid might make in trying to memorize the past or predict the future of a stimulus. These new analyses add insight to existing analyses based on mutual information; in particular, while channel capacity calculations reveal the noisiness of the estimator, they do not reveal its accuracy. Code is readily available. The technique extends beyond stimulus-dependent Maximum Entropy models.