As large tensor-variate data become increasingly common across applied machine learning and statistics, complex analysis methods for these data similarly increase in prevalence.  Such a trend offers the opportunity to understand subtler and more meaningful features of the data that, ostensibly, could not be studied with simpler datasets or simpler methodologies.  While promising, these advances are also perilous: novel analysis techniques do not always consider the possibility that their results are in fact an expected consequence of some simpler, already-known feature of simpler data.  For example, suppose one fits a time series model (e.g. Kalman Filter or multivariate GARCH) to data indexed by time, measurement dimension, and experimental sample.  Was a particular model fit achieved simply because the data was temporally smooth, and/or had correlated measurements (or samples)?  I will present two works that address this growing problem, the first of which uses Kronecker algebra to derive a tensor-variate maximum entropy distribution that has user-specified moments along each mode.  This distribution forms the basis of a statistical hypothesis test, and I will use this test to answer two active debates in the neuroscience community over the triviality of certain observed structure in data.  In the second part, I will discuss how to extend this maximum entropy formulation to arbitrary constraints using deep neural network architectures in the flavor of implicit generative modeling, and I will use this method in a texture synthesis application.