Multinomial Inverse Regression for Text Analysis

Text data, including speeches, stories, and other document forms, are often connected to sentiment variables that are of interest for research in marketing, economics, and elsewhere. It is also very high dimensional and difficult to incorporate into statistical analyses. This article introduces a straightforward framework of sentiment-sufficient dimension reduction for text data. Multinomial inverse regression is introduced as a general tool for simplifying predictor sets that can be represented as draws from a multinomial distribution, and we show that logistic regression of phrase counts onto document annotations can be used to obtain low-dimensional document representations that are rich in sentiment information. To facilitate this modeling, a novel estimation technique is developed for multinomial logistic regression with very high-dimensional response. In particular, independent Laplace priors with unknown variance are assigned to each regression coefficient, and we detail an efficient routine for maximization of the joint posterior over coefficients and their prior scale. This "gamma-lasso" scheme yields stable and effective estimation for general high-dimensional logistic regression, and we argue that it will be superior to current methods in many settings. Guidelines for prior specification are provided, algorithm convergence is detailed, and estimator properties are outlined from the perspective of the literature on nonconcave likelihood penalization. Related work on sentiment analysis from statistics, econometrics, and machine learning is surveyed and connected. Finally, the methods are applied in two detailed examples and we provide out-of-sample prediction studies to illustrate their effectiveness.