Asymptotic Theory for Linear-Chain Conditional Random Fields
Mathieu Sinn, Pascal Poupart · International Conference on Artificial Intelligence and Statistics · 2011
In this theoretical paper we develop an asymptotic theory for Linear-Chain Condi- tional Random Fields (L-CRFs) and apply it to derive conditions under which the Max- imum Likelihood Estimates (MLEs) of the model weights are strongly consistent. We first define L-CRFs for infinite sequences and analyze some of their basic properties. Then we establish conditions under which ergodic- ity of the observations implies ergodicity of the joint sequence of observations and labels. This result is the key ingredient to derive con- ditions for strong consistency of the MLEs. Interesting findings are that the consistency crucially depends on the limit behavior of the Hessian of the likelihood function and that, asymptotically, the state feature functions do not matter. In this paper, we study asymptotical properties of the Maximum Likelihood Estimates (MLEs) for the model weights. More specifically, we assume that we are given a sequence of observations and labels where the distribution of the labels conditional on the observa- tions follows an L-CRF with known feature functions but unknown weights. In this setting, we investigate conditions under which the MLEs converge to the true weights as the length of the sequences goes to infinity. Note that, to state and to analyze this problem, a def- inition of L-CRFs for infinite sequences is required. Our research is motivated by the following questions: How can the weights and feature functions be jointly estimated in the case where both are unknown? How robust is the training and inference towards a sampling bias (that is, when training and test data come from dierent distributions)? How well is the model iden- tifiable in the presence of noisy data? To tackle these problems of great practical importance, the present paper aims to achieve a better understanding of the simplest case, namely, when the feature functions are known and a sampling bias or noisy data is absent. Furthermore, it provides a theoretical framework and useful techniques to study the more complicated cases. This paper is structured as follows: In Sec. 2 we in- troduce some notation and review the definition of L- CRFs for finite sequences. In Sec. 3 we define L-CRFs for infinite sequences and derive some of their basic properties. Sec. 4 establishes conditions under which ergodicity of the sequence of observations implies er- godicity of the joint sequence of observations and la- bels. In Sec. 5 we apply the previous results to derive conditions under which the MLEs are strongly consis- tent. Sec. 6 concludes the paper.