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The Ising model was originally developed to model magnetisation of solids instatistical physics. As a network of binary variables with the probability ofbecoming 'active' depending only on direct neighbours, the Ising model appearsappropriate for many other processes. For instance, it was recently applied inpsychology to model co-occurrences of mental disorders. It has been shown thatthe connections between the variables (nodes) in the Ising network can beestimated with a series of logistic regressions. This naturally leads toquestions of how well such a model predicts new observations and how wellparameters of the Ising model can be estimated using logistic regressions. Herewe focus on the high-dimensional setting with more parameters than observationsand consider violations of assumptions of the lasso. In particular, wedetermine the consequences for both prediction and estimation when the sparsityand restricted eigenvalue assumptions are not satisfied. We explain by usingthe idea of connected copies (extreme multicollinearity) the fact thatprediction becomes better when either sparsity or multicollinearity is notsatisfied. We illustrate these results with simulations.

Logistic regression and Ising networks: prediction and estimation when violating lasso assumptions