A Kernelized Stein Discrepancy for Goodness-of-fit Tests

By Qiang Liu, Jason D. Lee, Michael I. Jordan. 2016

566 citations recorded by the source corpus.

We derive a new discrepancy statistic for measuring differences between two probability distributions based on combining Stein's identity with the reproducing kernel Hilbert space theory. We apply our result to test how well a probabilistic model fits a set of observations, and derive a new class of powerful goodness-of-fit tests that are widely applicable for complex and high dimensional distributions, even for those with computationally intractable normalization constants. Both theoretical and empirical properties of our methods are studied thoroughly.

Subfields: Statistics and Probability, Evaluation & Benchmarks

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Citations in the corpus

Generative Modeling by Estimating Gradients of the Data Distribution (2019, methodology): “In this paper, we explore a new principle for generative modeling based on estimating and sampling from the (Stein) score [33] of the logarithmic data density, which is the gradient of the log-density function at the input data point.”
Noise Estimation for Generative Diffusion Models (2021, methodology): “The proposed model estimates and samples the logarithm of the data density, which is the Stein score function [19].”
Kernel Mean Embedding of Distributions: A Review and Beyonds (2016, methodology): “Liu et al. (2016) and Chwialkowski et al. (2016) independently and simultaneously develop a goodness-of-fit test based on Stein’s identity applied to functions in RKHS (Oates et al. 2016).”
A Kernel Test of Goodness of Fit (2016, background): “An identical test was obtained simultaneously in independent work by Liu et al. (2016), for uncorrelated samples.”
Posterior Meta-Replay for Continual Learning (2021, background): “One option is to use a sample-based distance estimate between the distributions [16, 44], which would act upon the output of the WG network (i.e., the weight samples).”
Batch Stationary Distribution Estimation (2020, methodology): “A more relevant approach is Stein importance sampling (Liu and Lee, 2017), where the ratio is estimated by minimizing the kernelized Stein discrepancy (Liu et al., 2016).”
Adversarial Feature Matching for Text Generation (2017, methodology): “MMD, kernel-based discrepancy metrics such as kernelized Stein discrepancy (Liu et al., 2016; Wang & Liu, 2016) have been shown to be computationally tractable, while maintaining statistical power.”
Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm (2016, background): “smooth variable transforms and a recently introduced kernelized Stein discrepancy [10, 8, 9], which allows us to derive a closed form solution for the optimal smooth perturbation direction that gives the steepest descent on the KL divergence within the unit ball of a reproducing kernel Hilbert space”
Sliced Score Matching: A Scalable Approach to Density and Score Estimation (2019, background): “Li & Turner (2018) propose to set h ( x ) as the feature map of some kernel in the Stein class (Liu et al., 2016) of q , and solve a ﬁnite-sample version of (12) to obtain estimates of ∇ x log q ( x ) at the sample points.”
Learning Gradient Fields for Shape Generation (2020, neutral): “005 .”
A Kernel Loss for Solving the Bellman Equation (2019, methodology): “In fact, by using methods from kernel-based hypothesis testing (e.g., Gretton et al., 2012; Liu et al., 2016; Chwialkowski et al., 2016), one can design statistically calibrated methods to test if V = V π has been achieved, which may be useful for designing efﬁcient exploration strategies.”
Stein Neural Sampler (2018, unknown): “Fisher Divergence vs. KSD Fisher divergence dominates KSD in the following sense (Liu et al., 2016): Fisher divergence is stronger than KSD, and lot of other distances between distributions, such as total variation, Hellinger distance, Wasserstein distance, etc (Ley et al., 2013).”
Measuring Sample Quality with Diffusions (2016, methodology): “While our chief motivation is sample quality measurement, our work is also directly applicable to a variety of inferential tasks that currently rely on the Langevin operator, including control variate design [67, 68], one sample hypothesis testing [15, 58], variational inference [57], and importance”
Monte Carlo Gradient Estimation in Machine Learning (2019, background): “Stein’s identity (Gorham and Mackey, 2015; Liu et al., 2016) tells us that if a distribution p(x) is continuously differentiable then the gradient of its log-probability with respect to the random variable x can be related to the gradient of a set of test functions t(x) by the integral: Ep(x)…”
How to Train Your Energy-Based Models (2021, methodology): “With some regularity conditions (Gorham and Mackey, 2015; Liu et al., 2016), we have D Stein ( p data ( x ) (cid:107) p θ ( x )) ≥ 0 , where the equality holds if and only if p data ( x ) ≡ p θ ( x ) .”
Stein Variational Gradient Descent as Moment Matching (2018, background): “The readers are referred to Liu et al. (2016); Liu & Wang (2016); Liu et al. (2017) and references therein for more details.”
Functional Variational Bayesian Neural Networks (2019, background): “[14] Qiang Liu, Jason Lee, and Michael Jordan.”