Solving the 100 Swiss Francs Problem

Sturmfels offered 100 Swiss Francs in 2005 to a conjecture, which deals with a special case of the maximum likelihood estimation for a latent class model. This paper confirms the conjecture positively

β models for random hypergraphs with a given degree sequence

We introduce the beta model for random hypergraphs in order to represent the occurrence of multi-way interactions among agents in a social network. This model builds upon and generalizes the well-studied beta model for random graphs, which instead only considers pairwise interactions. We provide two algorithms for fitting the model parameters, IPS (iterative proportional scaling) and fixed point algorithm, prove that both algorithms converge if maximum likelihood estimator (MLE) exists, and provide algorithmic and geometric ways of dealing the issue of MLE existence

Multiple‐systems analysis for the quantification of modern slavery: classical and Bayesian approaches

Multiple systems estimation is a key approach for quantifying hidden populations such as the number of victims of modern slavery. The UK Government published an estimate of 10,000 to 13,000 victims, constructed by the present author, as part of the strategy leading to the Modern Slavery Act 2015. This estimate was obtained by a stepwise multiple systems method based on six lists. Further investigation shows that a small proportion of the possible models give rather different answers, and that other model fitting approaches may choose one of these. Three data sets collected in the field of modern slavery, together with a data set about the death toll in the Kosovo conflict, are used to investigate the stability and robustness of various multiple systems estimate methods. The crucial aspect is the way that interactions between lists are modelled, because these can substantially affect the results. Model selection and Bayesian approaches are considered in detail, in particular to assess their stability and robustness when applied to real modern slavery data. A new Markov Chain Monte Carlo Bayesian approach is developed; overall, this gives robust and stable results at least for the examples considered. The software and datasets are freely and publicly available to facilitate wider implementation and further research

Differentially Private Model Selection with Penalized and Constrained Likelihood

In statistical disclosure control, the goal of data analysis is twofold: The released information must provide accurate and useful statistics about the underlying population of interest, while minimizing the potential for an individual record to be identified. In recent years, the notion of differential privacy has received much attention in theoretical computer science, machine learning, and statistics. It provides a rigorous and strong notion of protection for individuals' sensitive information. A fundamental question is how to incorporate differential privacy into traditional statistical inference procedures. In this paper we study model selection in multivariate linear regression under the constraint of differential privacy. We show that model selection procedures based on penalized least squares or likelihood can be made differentially private by a combination of regularization and randomization, and propose two algorithms to do so. We show that our private procedures are consistent under essentially the same conditions as the corresponding non-private procedures. We also find that under differential privacy, the procedure becomes more sensitive to the tuning parameters. We illustrate and evaluate our method using simulation studies and two real data examples

Sharing Social Network Data: Differentially Private Estimation of Exponential-Family Random Graph Models

Motivated by a real-life problem of sharing social network data that contain sensitive personal information, we propose a novel approach to release and analyze synthetic graphs in order to protect privacy of individual relationships captured by the social network while maintaining the validity of statistical results. A case study using a version of the Enron e-mail corpus dataset demonstrates the application and usefulness of the proposed techniques in solving the challenging problem of maintaining privacy \emph{and} supporting open access to network data to ensure reproducibility of existing studies and discovering new scientific insights that can be obtained by analyzing such data. We use a simple yet effective randomized response mechanism to generate synthetic networks under $\epsilon$-edge differential privacy, and then use likelihood based inference for missing data and Markov chain Monte Carlo techniques to fit exponential-family random graph models to the generated synthetic networks.Comment: Updated, 39 page

Statistical Inference in a Directed Network Model with Covariates

Networks are often characterized by node heterogeneity for which nodes exhibit different degrees of interaction and link homophily for which nodes sharing common features tend to associate with each other. In this paper, we propose a new directed network model to capture the former via node-specific parametrization and the latter by incorporating covariates. In particular, this model quantifies the extent of heterogeneity in terms of outgoingness and incomingness of each node by different parameters, thus allowing the number of heterogeneity parameters to be twice the number of nodes. We study the maximum likelihood estimation of the model and establish the uniform consistency and asymptotic normality of the resulting estimators. Numerical studies demonstrate our theoretical findings and a data analysis confirms the usefulness of our model.Comment: 29 pages. minor revisio

Gravitational and electromagnetic fields of a charged tachyon

An axially symmetric exact solution of the Einstein-Maxwell equations is obtained and is interpreted to give the gravitational and electromagnetic fields of a charged tachyon. Switching off the charge parameter yields the solution for the uncharged tachyon which was earlier obtained by Vaidya. The null surfaces for the charged tachyon are discussed.Comment: 8 pages, LaTex, To appear in Pramana- J. Physic