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Many machine learning applications predict individual probabilities, such as the likelihood that a person develops a particular illness. Since these probabilities are unknown, a key question is how to address situations in which different models trained on the same dataset produce varying predictions for certain individuals. This issue is exemplified by the model multiplicity (MM) phenomenon, where a set of comparable models yield inconsistent predictions. Roth, Tolbert, and Weinstein recently introduced a reconciliation procedure, the Reconcile algorithm, to address this problem. Given two disagreeing models, the algorithm leverages their disagreement to falsify and improve at least one of the models. In this paper, we empirically analyze the Reconcile algorithm using five widely-used fairness datasets: COMPAS, Communities and Crime, Adult, Statlog (German Credit Data), and the ACS Dataset. We examine how Reconcile fits within the model multiplicity literature and compare it to existing MM solutions, demonstrating its effectiveness. We also discuss potential improvements to the Reconcile algorithm theoretically and practically. Finally, we extend the Reconcile algorithm to the setting of causal inference, given that different competing estimators can again disagree on specific causal average treatment effect (CATE) values. We present the first extension of the Reconcile algorithm in causal inference, analyze its theoretical properties, and conduct empirical tests. Our results confirm the practical effectiveness of Reconcile and its applicability across various domains., Comment: Presented at the ICML workshop on Humans, Algorithmic Decision-Making and Society: Modeling Interactions and Impact 2024

The class of recurrent mixture density networks is an important class of probabilistic models used extensively in sequence modeling and sequence-to-sequence mapping applications. In this class of models, the density of a target sequence in each time-step is modeled by a Gaussian mixture model with the parameters given by a recurrent neural network. In this paper, we generalize recurrent mixture density networks by defining a Gaussian mixture model on a non-linearly transformed target sequence in each time-step. The non-linearly transformed space is created by normalizing flow. We observed that this model significantly improves the fit to image sequences measured by the log-likelihood. We also applied the proposed model on some speech and image data, and observed that the model has significant modeling power outperforming other state-of-the-art methods in terms of the log-likelihood.