Corpus ID: 208512627

# Maximum likelihood estimation for discrete exponential families and random graphs

@article{Bogdan2019MaximumLE,
title={Maximum likelihood estimation for discrete exponential families and random graphs},
author={K. Bogdan and Michal Bosy and T. Skalski},
journal={arXiv: Probability},
year={2019}
}
• Published 2019
• Mathematics
• arXiv: Probability
We characterize the existence of the maximum likelihood estimator for discrete exponential families. Our criterion is simple to apply, as we show in various settings, most notably for exponential models of random graphs. As application we point out the size of independent identically distributed samples for which the maximum likelihood estimator exists with high probability.

#### References

SHOWING 1-10 OF 51 REFERENCES
On Existence of Maximum Likelihood Estimators in Exponential Families
• Mathematics
• 2000
We propose a simple necessary and sufficient condition for existence of maximum likelihood estimators in a large class of canonical exponential families. We give an application to log-spline families.
Maximum likelihood estimation in log-linear models
• Mathematics
• 2012
We study maximum likelihood estimation in log-linear models under conditional Poisson sampling schemes. We derive necessary and sufficient conditions for existence of the maximum likelihood estimatorExpand
On the geometry of discrete exponential families with application to exponential random graph models
• Mathematics
• 2008
There has been an explosion of interest in statistical models for analyzing network data, and considerable interest in the class of exponential random graph (ERG) models, especially in connectionExpand
On the expectation of the maximum of IID geometric random variables
A study of the expected value of the maximum of independent, identically distributed (IID) geometric random variables is presented based on the Fourier analysis of the distribution of the fractionalExpand
Graph model selection using maximum likelihood
• Computer Science, Mathematics
• ICML
• 2006
This work designs and implements MCMC algorithms for computing the maximum likelihood for four popular models: a power-law random graph model, a preferential attachment models, a small-world model, and a uniform random graph models, using Maximum Likelihood to compare graph models and select their parameters. Expand
Exponential-Family Models of Random Graphs: Inference in Finite-, Super-, and Infinite Population Scenarios
• Mathematics, Computer Science
• 2017
The core statistical notions of "sample" and "population" in the ERGM framework are clarified, and the process that generates the population graph from the observation process is separated, and likelihood-based inference in finite-, super-, and infinite-population scenarios are reviewed. Expand
Three tutorial lectures on entropy and counting
We explain the notion of the {\em entropy} of a discrete random variable, and derive some of its basic properties. We then show through examples how entropy can be useful as a combinatorialExpand
Estimating and understanding exponential random graph models
• Mathematics, Physics
• 2013
We introduce a method for the theoretical analysis of exponential random graph models. The method is based on a large-deviations approximation to the normalizing constant shown to be consistent usingExpand
Extreme point models in statistics
We give a survey of the general theory of extreme point models in statistics, i.e. statistical models that are given as the extreme points of the convex set of probability measures satisfying (in aExpand
Marginal Likelihoods for Distributed Parameter Estimation of Gaussian Graphical Models
• Mathematics, Computer Science
• IEEE Transactions on Signal Processing
• 2014
A general framework for distributed estimation based on a maximum marginal likelihood (MML) approach that computes local parameter estimates by maximizing marginal likelihoods defined with respect to data collected from local neighborhoods and is suitable for high-dimensional problems. Expand