Probability distributions produced by the cross-entropy loss for ordinal classification problems can possess undesired properties. We propose a straightforward technique to constrain discrete ordinal probability distributions to be unimodal via a combination of the Poisson probability mass function and the softmax nonlinearity. We evaluate this approach on two large ordinal image datasets and obtain promising results.
There are other ordinal techniques but which do not impose unimodal constraints. The proportional odds model (POM) and its neural network extensions (POMNN, CHNN @cite_2 ) do not suffer from the monotonicity issue due to the utilization of monotonically increasing biases in the calculation of probabilities. The stick-breaking approach by , which is a reformulation of the multinomial logit (softmax), could also be used in the ordinal case as it technically imposes an ordering on classes.
@cite_2
2067001387
This article describes the gologit2 program for generalized ordered logit models. gologit2 is inspired by Vincent Fu’s gologit routine (Stata Technical Bulletin Reprints 8: 160–164) and is backward compatible with it but offers several additional powerful options. A major strength of gologit2 is that it can fit three special cases of the generalized model: the proportional odds parallel-lines model, the partial proportional odds model, and the logistic regression model. Hence, gologit2 can fit models that are less restrictive than the parallel-lines models fitted by ologit (whose assumptions are often violated) but more parsimonious and interpretable than those fitted by a nonordinal method, such as multinomial logistic regression (i.e., mlogit). Other key advantages of gologit2 include support for linear constraints, survey data estimation, and the computation of estimated probabilities via the predict command. Threshold models are one of the most common approaches for ordinal regression, based on projecting patterns to the real line and dividing this real line in consecutive intervals, one interval for each class. However, finding such one-dimensional projection can be too harsh an imposition for some datasets. This paper proposes a multidimensional latent space representation with the purpose of relaxing this projection, where the different classes are arranged based on concentric hyperspheres, each class containing the previous classes in the ordinal scale. The proposal is implemented through a neural network model, each dimension being a linear combination of a common set of basis functions. The model is compared to a nominal neural network, a neural network based on the proportional odds model and to other state-of-the-art ordinal regression methods for a total of 12 datasets. The proposed latent space shows an improvement on the two performance metrics considered, and the model based on the three-dimensional latent space obtains competitive performance when compared to the other methods. The development of accurate models and ecient algorithms for the analysis of multivariate categorical data are important and longstanding problems in machine learning and computational statistics. In this paper, we focus on modeling categorical data using Latent Gaussian Models (LGMs). We propose a novel stick-breaking likelihood function for categorical LGMs that exploits accurate linear and quadratic bounds on the logistic log-partition function, leading to an eective variational inference and learning framework. We thoroughly compare our approach to existing algorithms for multinomial logit probit likelihoods on several problems, including inference in multinomial Gaussian process classification and learning in latent factor models. Our extensive comparisons demonstrate that our stick-breaking model eectively captures correlation in discrete data and is well suited for the analysis of categorical data. Discrete choice models generally assume that model specification is known a priori. In practice, determining the utility specification for a particular application remains a difficult task and model misspecification may lead to biased parameter estimates. In this paper, we propose a new mathematical framework for estimating choice models in which the systematic part of the utility specification is divided into an interpretable part and a learning representation part that aims at automatically discovering a good utility specification from available data. We show the effectiveness of our framework by augmenting the utility specification of the Multinomial Logit Model (MNL) with a new non-linear representation arising from a Neural Network (NN). This leads to a new choice model referred to as the Learning Multinomial Logit (L-MNL) model. Our experiments show that our L-MNL model outperformed the traditional MNL models and existing hybrid neural network models both in terms of predictive performance and accuracy in parameter estimation.