Revisiting Classifier Two-Sample Tests (1610.06545v4)

Abstract: The goal of two-sample tests is to assess whether two samples, $S_P \sim P^n$ and $S_Q \sim Q^m$, are drawn from the same distribution. Perhaps intriguingly, one relatively unexplored method to build two-sample tests is the use of binary classifiers. In particular, construct a dataset by pairing the $n$ examples in $S_P$ with a positive label, and by pairing the $m$ examples in $S_Q$ with a negative label. If the null hypothesis "$P = Q$" is true, then the classification accuracy of a binary classifier on a held-out subset of this dataset should remain near chance-level. As we will show, such Classifier Two-Sample Tests (C2ST) learn a suitable representation of the data on the fly, return test statistics in interpretable units, have a simple null distribution, and their predictive uncertainty allow to interpret where $P$ and $Q$ differ. The goal of this paper is to establish the properties, performance, and uses of C2ST. First, we analyze their main theoretical properties. Second, we compare their performance against a variety of state-of-the-art alternatives. Third, we propose their use to evaluate the sample quality of generative models with intractable likelihoods, such as Generative Adversarial Networks (GANs). Fourth, we showcase the novel application of GANs together with C2ST for causal discovery.

• The paper introduces C2ST, a binary classifier framework to test if two samples originate from the same distribution.
• It establishes rigorous theoretical foundations with asymptotic analyses and empirical evaluations against methods like MMD and ME tests.
• The study applies C2ST to generative model evaluation and causal discovery, showcasing its scalability and interpretability in complex models.

An Examination of Classifier Two-Sample Tests

The paper "Revisiting Classifier Two-Sample Tests" by David Lopez-Paz and Maxime Oquab presents an insightful analysis of using binary classifiers for two-sample testing. Two-sample tests are pivotal in statistical analysis to determine if two datasets originate from the same distribution, denoted as hypothesis $H_0: P = Q$. This methodology is paramount in assessing generative models, particularly intractable ones like GANs, and this paper explores the classifier-driven approach to enhance the efficacy of two-sample tests.

Overview

The authors propose using a binary classifier to distinguish between two samples by labeling the first sample with one class and the second with another. If the classifier's accuracy is close to chance, it suggests the null hypothesis cannot be rejected, supporting the case that both samples are from the same distribution. In contrast, significant deviation from this baseline would indicate that the two samples originate from different distributions. This approach, termed as Classifier Two-Sample Tests (C2ST), adapts on-the-fly representations and uses interpretable units for test statistics.

Key Contributions

1. Theoretical Underpinning: The paper rigorously establishes the theoretical underpinnings of C2ST, detailing asymptotic distributions under both null and alternative hypotheses. The authors provide mathematical formulations to derive the testing power and accurately estimate p-values.
2. Performance Evaluation: Extensive empirical evaluation against several state-of-the-art methods is conducted on both synthetic and real-world datasets. Notable results show that C2ST performs competitively, often surpassing traditional methods like the Maximum Mean Discrepancy (MMD) and ME tests.
3. Generative Model Evaluation: A significant application is evaluating the fidelity of generated samples from GANs. The authors aptly demonstrate that C2ST can robustly identify the discrepancies between real and generated data, aiding in tuning generative models.
4. Novel Application in Causal Discovery: The paper extends C2ST's usability to causal discovery, utilizing Conditional GANs to capture cause-effect relationships between variables beyond the restrictive additive noise models without relying on independence assumptions.

Implications and Future Directions

The C2ST framework offers several practical and theoretical advancements. Practically, it provides a scalable and interpretable method for evaluating complex models where traditional testing methods might fall short. Theoretically, it lays the groundwork for exploring higher-order two-sample statistics and adapting deep learning paradigms to classical statistical testing environments.

The paper hints at further research in optimizing the classifier's choice and structure, which presents an open frontier in both the statistical and AI disciplines. Future endeavors could investigate advanced feature interpretations and extend C2ST methodologies to unsupervised or semi-supervised contexts, potentially revolutionizing two-sample testing paradigms within AI research.

In summary, this work stands as a testament to the fruitful intersection of machine learning and statistical practices, inspiring further cross-disciplinary explorations. The C2ST method not only reinforces the efficiency of classifier utilization in statistical testing but also opens avenues for innovative applications and model evaluations within AI.