Statistical Inference for Generative Models with Maximum Mean Discrepancy (1906.05944v1)

Abstract: While likelihood-based inference and its variants provide a statistically efficient and widely applicable approach to parametric inference, their application to models involving intractable likelihoods poses challenges. In this work, we study a class of minimum distance estimators for intractable generative models, that is, statistical models for which the likelihood is intractable, but simulation is cheap. The distance considered, maximum mean discrepancy (MMD), is defined through the embedding of probability measures into a reproducing kernel Hilbert space. We study the theoretical properties of these estimators, showing that they are consistent, asymptotically normal and robust to model misspecification. A main advantage of these estimators is the flexibility offered by the choice of kernel, which can be used to trade-off statistical efficiency and robustness. On the algorithmic side, we study the geometry induced by MMD on the parameter space and use this to introduce a novel natural gradient descent-like algorithm for efficient implementation of these estimators. We illustrate the relevance of our theoretical results on several classes of models including a discrete-time latent Markov process and two multivariate stochastic differential equation models.

• The paper introduces MMD-based statistical estimators for generative models, establishing their consistency and asymptotic normality.
• It presents a novel natural gradient descent algorithm that boosts computational efficiency in high-dimensional inference.
• The robust MMD framework effectively handles model misspecification, making it suitable for applications like GANs and variational autoencoders.

Insights into Statistical Inference for Generative Models Using Maximum Mean Discrepancy

The paper "Statistical Inference for Generative Models with Maximum Mean Discrepancy" by François-Xavier Briol, Alessandro Barp, Andrew B. Duncan, and Mark Girolami offers a comprehensive examination of statistical estimators for generative models, where likelihoods are intractable but simulation is feasible. The focus is on leveraging the Maximum Mean Discrepancy (MMD) as a divergence criterion for inference, exploring theoretical aspects and proposing efficient computational strategies for determining model parameters.

Exploration of MMD as a Divergence for Inference

Generative models are vital across various domains, but the complexity and intractability of their likelihoods pose challenges. This work proposes using MMD, a kernel-based metric, to provide a flexible, robust alternative for inference in these scenarios. MMD is a powerful tool due to its ability to measure discrepancies between empirical and parametric distributions effectively. By embedding probability measures into a reproducing kernel Hilbert space, MMD provides a framework in which statistical properties can be explored.

Theoretical Foundations and Estimator Properties

The paper establishes the theoretical properties of MMD-based estimators, demonstrating their consistency and asymptotic normality in well-specified settings (M-closed). It also showcases robustness to model misspecification in open settings (M-open), which is crucial when dealing with real-world data. The flexibility in choosing kernels allows researchers to balance between statistical efficiency and robustness according to specific application needs.

Computational Advancements: Natural Gradient Descent

The authors present a novel algorithm inspired by natural gradient descent, tailored to exploit the geometry induced by MMD on parameter spaces. This approach significantly enhances computational efficiency, particularly in high-dimensional settings or when dealing with costly simulations. This method aligns the optimization process with the unique information geometry of MMD, providing practical benefits in terms of speed and convergence.

Implications and Practical Applications

This work has notable implications in the field of complex generative models, which are prevalent in machine learning areas such as GANs and variational autoencoders. The robustness against misspecification makes MMD-based approaches suitable for applications involving synthetic data generation, understanding ecological systems, and modeling dynamic processes in finance and biology.

Future Directions and Research Opportunities

The paper opens several avenues for future research, including refining the computational cost of MMD estimation in large-scale data settings, exploring the role of kernel choice in diverse applications, and extending methodologies to include optimization capabilities in model reduction scenarios. Furthermore, the robustness properties of MMD estimators warrant deeper exploration, especially in adaptive settings where robustness to unseen data or environmental changes could be pivotal.

Conclusion

Overall, this paper provides valuable insights and offers rigorous foundations and practical methodologies for applying MMD in statistical inference for generative models. Its contribution is significant in advancing the capability to perform robust and efficient inference without the need for complex and often inaccessible likelihood calculations, making it a critical reference point for researchers tackling intractable generative models across domains.