Mathematics

The p -tensor Ising model is a one-parameter discrete exponential family for modeling dependent binary data, where the sufficient statistic is a multi-linear form of degree p≥2 . This is a natural generalization of the matrix Ising model, that provides a convenient mathematical framework for capturing higher-order dependencies in complex relational data. In this paper, we consider the problem of estimating the natural parameter of the p -tensor Ising model given a single sample from the distribution on N nodes. Our estimate is based on the maximum pseudo-likelihood (MPL) method, which provides a computationally efficient algorithm for estimating the parameter that avoids computing the intractable partition function. We derive general conditions under which the MPL estimate is N − − √ -consistent, that is, it converges to the true parameter at rate 1/ N − − √ . In particular, we show the N − − √ -consistency of the MPL estimate in the p -spin Sherrington-Kirkpatrick (SK) model, spin systems on general p -uniform hypergraphs, and Ising models on the hypergraph stochastic block model (HSBM). In fact, for the HSBM we pin down the exact location of the phase transition threshold, which is determined by the positivity of a certain mean-field variational problem, such that above this threshold the MPL estimate is N − − √ -consistent, while below the threshold no estimator is consistent. Finally, we derive the precise fluctuations of the MPL estimate in the special case of the p -tensor Curie-Weiss model. An interesting consequence of our results is that the MPL estimate in the Curie-Weiss model saturates the Cramer-Rao lower bound at all points above the estimation threshold, that is, the MPL estimate incurs no loss in asymptotic efficiency, even though it is obtained by minimizing only an approximation of the true likelihood function for computational tractability.

In a multi-index model with k index vectors, the input variables are transformed by taking inner products with the index vectors. A transfer function f: R k →R is applied to these inner products to generate the output. Thus, multi-index models are a generalization of linear models. In this paper, we consider monotone multi-index models. Namely, the transfer function is assumed to be coordinate-wise monotone. The monotone multi-index model therefore generalizes both linear regression and isotonic regression, which is the estimation of a coordinate-wise monotone function. We consider the case of nonnegative index vectors. We provide an algorithm based on integer programming for the estimation of monotone multi-index models, and provide guarantees on the L 2 loss of the estimated function relative to the ground truth.

This article studies the estimation of static community memberships from temporally correlated pair interactions represented by an N -by- N -by- T tensor where N is the number of nodes and T is the length of the time horizon. We present several estimation algorithms, both offline and online, which fully utilise the temporal nature of the observed data. As an information-theoretic benchmark, we study data sets generated by a dynamic stochastic block model, and derive fundamental information criteria for the recoverability of the community memberships as N→∞ both for bounded and diverging T . These results show that (i) even a small increase in T may have a big impact on the recoverability of community memberships, (ii) consistent recovery is possible even for very sparse data (e.g. bounded average degree) when T is large enough. We analyse the accuracy of the proposed estimation algorithms under various assumptions on data sparsity and identifiability, and prove that an efficient online algorithm is strongly consistent up to the information-theoretic threshold under suitable initialisation. Numerical experiments show that even a poor initial estimate (e.g., blind random guess) of the community assignment leads to high accuracy after a small number of iterations, and remarkably so also in very sparse regimes.

Consider bivariate observations ( X 1 , Y 1 ),…,( X n , Y n )∈R×R with unknown conditional distributions Q x of Y , given that X=x . The goal is to estimate these distributions under the sole assumption that Q x is isotonic in x with respect to likelihood ratio order. If the observations are identically distributed, a related goal is to estimate the joint distribution L(X,Y) under the sole assumption that it is totally positive of order two in a certain sense. After reviewing and generalizing the concepts of likelihood ratio order and total positivity of order two, an algorithm is developed which estimates the unknown family of distributions ( Q x ) x via empirical likelihood. The benefit of the stronger regularization imposed by likelihood ratio order over the usual stochastic order is evaluated in terms of estimation and predictive performances on simulated as well as real data.

Assuming that a reflected Ornstein-Uhlenbeck state process is observed at discrete time instants, we propose generalized moment estimators to estimate all drift and diffusion parameters via the celebrated ergodic theorem. With the sampling time step h > 0 arbitrarily fixed, we prove the strong consistency and asymptotic normality of our estimators as the sampling size n tends to infinity. This provides a complete solution to an open problem left in Hu et al. [5].

We are interested in recovering information on a stochastic block model from the subgraph discovered by an exploring random walk. Stochastic block models correspond to populations structured into a finite number of types, where two individuals are connected by an edge independently from the other pairs and with a probability depending on their types. We consider here the dense case where the random network can be approximated by a graphon. This problem is motivated from the study of chain-referral surveys where each interviewee provides information on her/his contacts in the social network. First, we write the likelihood of the subgraph discovered by the random walk: biases are appearing since hubs and majority types are more likely to be sampled. Even for the case where the types are observed, the maximum likelihood estimator is not explicit any more. When the types of the vertices is unobserved, we use an SAEM algorithm to maximize the likelihood. Second, we propose a different estimation strategy using new results by Athreya and Roellin. It consists in de-biasing the maximum likelihood estimator proposed in Daudin et al. and that ignores the biases.

The marginal likelihood or evidence in Bayesian statistics contains an intrinsic penalty for larger model sizes and is a fundamental quantity in Bayesian model comparison. Over the past two decades, there has been steadily increasing activity to understand the nature of this penalty in singular statistical models, building on pioneering work by Sumio Watanabe. Unlike regular models where the Bayesian information criterion (BIC) encapsulates a first-order expansion of the logarithm of the marginal likelihood, parameter counting gets trickier in singular models where a quantity called the real log canonical threshold (RLCT) summarizes the effective model dimensionality. In this article, we offer a probabilistic treatment to recover non-asymptotic versions of established evidence bounds as well as prove a new result based on the Gibbs variational inequality. In particular, we show that mean-field variational inference correctly recovers the RLCT for any singular model in its canonical or normal form. We additionally exhibit sharpness of our bound by analyzing the dynamics of a general purpose coordinate ascent algorithm (CAVI) popularly employed in variational inference.

The study of records in the Linear Drift Model (LDM) has attracted much attention recently due to applications in several fields. In the present paper we study δ -records in the LDM, defined as observations which are greater than all previous observations, plus a fixed real quantity δ . We give analytical properties of the probability of δ -records and study the correlation between δ -record events. We also analyse the asymptotic behaviour of the number of δ -records among the first n observations and give conditions for convergence to the Gaussian distribution. As a consequence of our results, we solve a conjecture posed in J. Stat. Mech. 2010, P10013, regarding the total number of records in a LDM with negative drift. Examples of application to particular distributions, such as Gumbel or Pareto are also provided. We illustrate our results with a real data set of summer temperatures in Spain, where the LDM is consistent with the global-warming phenomenon.

It is clear that conventional statistical inference protocols need to be revised to deal correctly with the high-dimensional data that are now common. Most recent studies aimed at achieving this revision rely on powerful approximation techniques, that call for rigorous results against which they can be tested. In this context, the simplest case of high-dimensional linear regression has acquired significant new relevance and attention. In this paper we use the statistical physics perspective on inference to derive a number of new exact results for linear regression in the high-dimensional regime.

We explore the class of exchangeable Bernoulli distributions building on their geometrical structure. Exchangeable Bernoulli probability mass functions are points in a convex polytope and we have found analytical expressions for their extremal generators. The geometrical structure turns out to be crucial to simulate high dimensional and negatively correlated binary data. Furthermore, for a wide class of statistical indices and measures of a probability mass function we are able to find not only their sharp bounds in the class, but also their distribution across the class. Estimate and testing are also addressed.