The Poisson transform for unnormalised statistical models


Statistics and Computing

June Forthcoming

Departments: Economics & Decision Sciences

Contrary to standard statistical models, unnormalised statistical models only specify the likelihood function up to a constant. While such models are natural and popular, the lack of normalisation makes inference much more difficult. Extending classical results on the multinomial-Poisson transform (Baker In: J Royal Stat Soc 43(4):495–504, 1994), we show that inferring the parameters of a unnormalised model on a space (Formula presented.) can be mapped onto an equivalent problem of estimating the intensity of a Poisson point process on (Formula presented.). The unnormalised statistical model now specifies an intensity function that does not need to be normalised. Effectively, the normalisation constant may now be inferred as just another parameter, at no loss of information. The result can be extended to cover non-IID models, which includes for example unnormalised models for sequences of graphs (dynamical graphs), or for sequences of binary vectors. As a consequence, we prove that unnormalised parameteric inference in non-IID models can be turned into a semi-parametric estimation problem. Moreover, we show that the noise-contrastive estimation method of Gutmann and Hyvärinen (J Mach Learn Res 13(1):307–361, 2012) can be understood as an approximation of the Poisson transform, and extended to non-IID settings. We use our results to fit spatial Markov chain models of eye movements, where the Poisson transform allows us to turn a highly non-standard model into vanilla semi-parametric logistic regression. © 2015 Springer Science+Business Media New York

A Comparative Overview of EU and US Legislative and Regulatory Systems


Columbia Journal of European Law


Departments: Tax & Law, GREGHEC (CNRS)

The aim of this report is to inform the EU-US Transatlantic Trade and Investment negotiations on enhanced regulatory coherence and cooperation, by providing negotiators, stakeholders and the public with a comparative overview of the US and EU legislative and regulatory processes in their current form, highlighting differences and similarities

A Legal Analysis of Packaging Standardisation Requirements Under EU Law - The Case of ‘Plain Packaging’ in the United Kingdom


Journal of Business Law


Departments: Tax & Law, GREGHEC (CNRS)

A Mathematical Turn in Business Regulation: The Rise of Legal Indicators


International Journal of Law in Context


Departments: Tax & Law

A note on 'Sourcing decisions with stochastic reliability and stochastic demand'


Production and Operations Management


Departments: Operations Management & Information Technology, GREGHEC (CNRS)

Keywords: sourcing;supplier selection;random yield

This note complements the study of Burke, Carillo, and Vakharia (2009 hereafter “BCV”) which analyzes a class of single-product multisourcing problems under stochastic demand and random yields. The purpose is twofold. First, we prove that the objective function used by these authors is only a lower bound for the expected profit for which we provide the correct expression. Second, we show on some of the numerical instances provided in BCV's study that the structure and the performance of the BCV ordering policy may be substantially different from the optimal ordering policy. We conclude by giving general qualitative insights characterizing suboptimality of the BCV solution