Improving Inference of Gaussian Mixtures Using Auxiliary Variables

Expanding a lower-dimensional problem to a higher-dimensional space and then projecting back is often beneficial. This article rigorously investigates this perspective in the context of finite mixture models, namely how to improve inference for mixture models by using auxiliary variables. Despite the large literature in mixture models and several empirical examples, there is no previous work that gives general theoretical justification for including auxiliary variables in mixture models, even for special cases. We provide a theoretical basis for comparing inference for mixture multivariate models with the corresponding inference for marginal univariate mixture models. Analytical results for several special cases are established. We show that the probability of correctly allocating mixture memberships and the information number for the means of the primary outcome in a bivariate model with two Gaussian mixtures are generally larger than those in each univariate model. Simulations under a range of scenarios, including misspecified models, are conducted to examine the improvement. The method is illustrated by two real applications in ecology and causal inference

An information criterion for auxiliary variable selection in incomplete data analysis

Statistical inference is considered for variables of interest, called primary variables, when auxiliary variables are observed along with the primary variables. We consider the setting of incomplete data analysis, where some primary variables are not observed. Utilizing a parametric model of joint distribution of primary and auxiliary variables, it is possible to improve the estimation of parametric model for the primary variables when the auxiliary variables are closely related to the primary variables. However, the estimation accuracy reduces when the auxiliary variables are irrelevant to the primary variables. For selecting useful auxiliary variables, we formulate the problem as model selection, and propose an information criterion for predicting primary variables by leveraging auxiliary variables. The proposed information criterion is an asymptotically unbiased estimator of the Kullback-Leibler divergence for complete data of primary variables under some reasonable conditions. We also clarify an asymptotic equivalence between the proposed information criterion and a variant of leave-one-out cross validation. Performance of our method is demonstrated via a simulation study and a real data example

Predictive Modelling of Retail Banking Transactions for Credit Scoring, Cross-Selling and Payment Pattern Discovery

Evaluating transactional payment behaviour offers a competitive advantage in the modern payment ecosystem, not only for confirming the presence of good credit applicants or unlocking the cross-selling potential between the respective product and service portfolios of financial institutions, but also to rule out bad credit applicants precisely in transactional payments streams. In a diagnostic test for analysing the payment behaviour, I have used a hybrid approach comprising a combination of supervised and unsupervised learning algorithms to discover behavioural patterns. Supervised learning algorithms can compute a range of credit scores and cross-sell candidates, although the applied methods only discover limited behavioural patterns across the payment streams. Moreover, the performance of the applied supervised learning algorithms varies across the different data models and their optimisation is inversely related to the pre-processed dataset. Subsequently, the research experiments conducted suggest that the Two-Class Decision Forest is an effective algorithm to determine both the cross-sell candidates and creditworthiness of their customers. In addition, a deep-learning model using neural network has been considered with a meaningful interpretation of future payment behaviour through categorised payment transactions, in particular by providing additional deep insights through graph-based visualisations. However, the research shows that unsupervised learning algorithms play a central role in evaluating the transactional payment behaviour of customers to discover associations using market basket analysis based on previous payment transactions, finding the frequent transactions categories, and developing interesting rules when each transaction category is performed on the same payment stream. Current research also reveals that the transactional payment behaviour analysis is multifaceted in the financial industry for assessing the diagnostic ability of promotion candidates and classifying bad credit applicants from among the entire customer base. The developed predictive models can also be commonly used to estimate the credit risk of any credit applicant based on his/her transactional payment behaviour profile, combined with deep insights from the categorised payment transactions analysis. The research study provides a full review of the performance characteristic results from different developed data models. Thus, the demonstrated data science approach is a possible proof of how machine learning models can be turned into cost-sensitive data models

Improving inference of Gaussian mixtures using auxiliary variables

Expanding a lower-dimensional problem to a higher-dimensional space and then projecting back is often beneficial. This article rigorously investigates this perspective in the context of finite mixture models, namely how to improve inference for mixture models by using auxiliary variables. Despite the large literature in mixture models and several empirical examples, there is no previous work that gives general theoretical justification for including auxiliary variables in mixture models, even for special cases. We provide a theoretical basis for comparing inference for mixture multivariate models with the corresponding inference for marginal univariate mixture models. Analytical results for several special cases are established. We show that the probability of correctly allocating mixture memberships and the information number for the means of the primary outcome in a bivariate model with two Gaussian mixtures are generally larger than those in each univariate model. Simulations under a range of scenarios, including misspecified models, are conducted to examine the improvement. The method is illustrated by two real applications in ecology and causal inference