The Poisson transform for unnormalised statistical models

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. Here we show that inferring the parameters of a unnormalised model on a space $\Omega$ can be mapped onto an equivalent problem of estimating the intensity of a Poisson point process on $\Omega$. 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 divergence of Gutmann & Hyv\"arinen (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

A Batch Noise Contrastive Estimation Approach for Training Large Vocabulary Language Models

Training large vocabulary Neural Network Language Models (NNLMs) is a difficult task due to the explicit requirement of the output layer normalization, which typically involves the evaluation of the full softmax function over the complete vocabulary. This paper proposes a Batch Noise Contrastive Estimation (B-NCE) approach to alleviate this problem. This is achieved by reducing the vocabulary, at each time step, to the target words in the batch and then replacing the softmax by the noise contrastive estimation approach, where these words play the role of targets and noise samples at the same time. In doing so, the proposed approach can be fully formulated and implemented using optimal dense matrix operations. Applying B-NCE to train different NNLMs on the Large Text Compression Benchmark (LTCB) and the One Billion Word Benchmark (OBWB) shows a significant reduction of the training time with no noticeable degradation of the models performance. This paper also presents a new baseline comparative study of different standard NNLMs on the large OBWB on a single Titan-X GPU.Comment: Accepted for publication at INTERSPEECH'1

Learning neural trans-dimensional random field language models with noise-contrastive estimation

Trans-dimensional random field language models (TRF LMs) where sentences are modeled as a collection of random fields, have shown close performance with LSTM LMs in speech recognition and are computationally more efficient in inference. However, the training efficiency of neural TRF LMs is not satisfactory, which limits the scalability of TRF LMs on large training corpus. In this paper, several techniques on both model formulation and parameter estimation are proposed to improve the training efficiency and the performance of neural TRF LMs. First, TRFs are reformulated in the form of exponential tilting of a reference distribution. Second, noise-contrastive estimation (NCE) is introduced to jointly estimate the model parameters and normalization constants. Third, we extend the neural TRF LMs by marrying the deep convolutional neural network (CNN) and the bidirectional LSTM into the potential function to extract the deep hierarchical features and bidirectionally sequential features. Utilizing all the above techniques enables the successful and efficient training of neural TRF LMs on a 40x larger training set with only 1/3 training time and further reduces the WER with relative reduction of 4.7% on top of a strong LSTM LM baseline.Comment: 5 pages and 2 figure

Statistical applications of contrastive learning

The likelihood function plays a crucial role in statistical inference and experimental design. However, it is computationally intractable for several important classes of statistical models, including energy-based models and simulator-based models. Contrastive learning is an intuitive and computationally feasible alternative to likelihood-based learning. We here first provide an introduction to contrastive learning and then show how we can use it to derive methods for diverse statistical problems, namely parameter estimation for energy-based models, Bayesian inference for simulator-based models, as well as experimental design.Comment: Accepted to Behaviormetrik