stat.MLMar 8, 2026

Probabilistic Inference and Learning with Stein's Method

Authors: Qiang Liu, Lester Mackey, Chris Oates

Organizations: University of Texas at Austin · Microsoft Research · Newcastle University · The Alan Turing Institute

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Abstract

This monograph provides a rigorous overview of theoretical and methodological aspects of probabilistic inference and learning with Stein's method. Recipes are provided for constructing Stein discrepancies from Stein operators and Stein sets, and properties of these discrepancies such as computability, separation, convergence detection, and convergence control are discussed. Further, the connection between Stein operators and Stein variational gradient descent is set out in detail. The main definitions and results are precisely stated, and references to all proofs are provided.

Figures & tables

≜\triangleqdefined as being equal
[n][n]{1,…,n}\{1,\dots,n\}
([n]m)\binom{[n]}{m}{σ⊆[n]:∣σ∣=m}\{\sigma\subseteq[n]:|\sigma|=m\} , all size mm subsets of [n][n]
∥⋅∥\|\cdot\|a generic norm on Rd\mathbb{R}^{d}
IS\mathbb{I}_{S}indicator function IS(x)=1\mathbb{I}_{S}(x)=1 if x∈Sx\in S , else IS(x)=0\mathbb{I}_{S}(x)=0
P\mathcal{P}the set of all probability measures on a measurable space that will be clear from context
∥⋅∥p\|\cdot\|_{p}the norm ∥x∥p≜(∣x1∣p+⋯+∣xd∣p)1/p\|x\|_{p}\triangleq(|x_{1}|^{p}+\dots+|x_{d}|^{p})^{1/p} on Rd\mathbb{R}^{d} , p∈[1,∞)p\in[1,\infty)
Table 1
Figure 1: Samples generated using the unadjusted Langevin algorithm (ULA). Here the target PP is a Gaussian mixture model with density pp and ϵ\epsilon denotes the step size parameter of ULA. Contours of log⁡p\log p are depicted.
Figure 2: Tuning the unadjusted Langevin algorithm (ULA). The left panel presents the mean square error associated with the empirical estimates for the means and variances of PP (which cannot be computed in general), while the right panel presents the kernel Stein discrepancy ( KSD ) (which can be computed). Here we used the inverse multi-quadric kernel ( Example 2.13 ) with exponent β=1\beta=1 and bandwidth ℓ=0.01\ell=0.01 . The experiment was repeated 10 times and means and standard errors are reported.
Figure 3: Goodness-of-fit testing with kernel Stein discrepancy. Here we plot the null distribution of the kernel Stein discrepancy test statistic as approximated using the wild bootstrap (blue), the actual value of the kernel Stein discrepancy test statistic (black), and the values for which the null would be rejected (shaded). The amount of model misspecification is denoted with σ\sigma , so that when σ=0\sigma=0 the model is correct. The inverse multi-quadric kernel with length scale ℓ\ell was used.
Figure 4: Goodness-of-fit testing with kernel Stein discrepancy. Here we plot the test power (i.e. the probability of rejecting the null) as a function of both the amount of model misspecification σ\sigma and the kernel length scale ℓ\ell . Means and standard errors over 100 experiments are reported.
Figure 5: Stein points (left) and Frank–Wolfe–Stein points (right) were used to approximate the Rosenbrock target ( 70 ) (black shaded).
Figure 6: Stein variational gradient descent with fixed kernel length scale ℓ\ell (left) and adaptively selected kernel length scale ℓ\ell (right) were used to approximate the Rosenbrock target ( 70 ) (black shaded).
Figure 7: Stein importance sampling with positive weights (left) and unconstrained weights (right) were used to approximate the Rosenbrock target ( 70 ) (black shaded). The size of each marker is proportional to the weight that the associated state is assigned; blue circles indicate a positive weight and red triangles indicate a negative weight.
Figure 8: Images generated by SteinGAN trained on the LSUN dataset ( Yu et al., 2015 ) , consisting of nearly 3M images of bedrooms. Reproduced with permission from Liu and Wang (2017) .
Figure 9: A comparison of variational approximations to a target PP (left) produced by minimizing kernel Stein discrepancy (centre) and Kullback–Leibler divergence (right). The variational approximation was based on a block neural autoregressive flow and the target PP arose from a Bayesian analysis of a biochemical oxygen demand model. Reproduced with permission from Fisher et al. (2021) .
Figure 10: Training variational autoencoders on the Fashion MNIST dataset; a binary latent zz of dimension d′=200d^{\prime}=200 was employed and m=2m=2 samples were used to evaluate each gradient. Reproduced with permission from Shi et al. (2022b) .
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