Positive and unlabelled learning is an important non-standard inference problem which arises naturally in many applications. The significant limitation of almost all existing methods addressing it lies in assuming that the propensity score function is constant and does not depend on features (Selected Completely at Random assumption), which is unrealistic in many practical situations. Avoiding this assumption, we consider parametric approach to the problem of joint estimation of posterior probability and propensity score functions. We show that if both these functions are logistic with different parameters (double logistic model) then the corresponding parameters are identifiable. Motivated by this, we propose two approaches to their estimation: a joint maximum likelihood method and the second approach based on an alternating maximization of two Fisher consistent approximations. Our experimental results show that the proposed methods perform on par or better than the existing methods based on Expectation-Maximisation scheme.

We consider a model selection problem in high-dimensional binary Markov random fields. The usefulness of the Ising model in studying systems of complex interactions has been confirmed in many papers. The main drawback of this model is the intractable norming constant that makes estimation of parameters very challenging. In the paper we propose a Lasso penalized version of the Monte Carlo maximum likelihood method. We prove that our algorithm, under mild regularity conditions, recognizes the true dependence structure of the graph with high probability. The efficiency of the proposed method is also investigated via numerical studies.

This note corrects an error in the proof of corollary 1 of Li et al. ( 2014 ). The original claim of the contraction principle in appendix D of Li et al. no longer holds.