Polynomial Stein Operators

• Polynomial Stein operators are linear differential operators with polynomial coefficients that characterize probability distributions via integration by parts identities.
• They are embedded in the first Weyl algebra, facilitating algorithmic construction and combinatorial analysis of high-order operators for complex random variable transformations.
• Applications include constructing discrepancy measures for Bayesian inference and characterizing distributions from sums, products, and Hermite polynomial transformations.

A polynomial Stein operator is a linear differential operator with polynomial coefficients that characterizes a target probability distribution via integration by parts identities. Such operators play a central role in extensions of Stein's method, particularly for analyzing complex distributions constructed from sums, products, and polynomial transformations of random variables. Their study involves operator algebras, noncommutative algebraic structures, and computational methods that link probability, analysis, and algebraic geometry.

1. Definition and Formalization

Let $W$ be a real random variable with smooth, everywhere non-vanishing density $p_W$ on an interval $J \subseteq \mathbb{R}$, and let $\mathcal{F}$ denote a suitable class of smooth test functions (e.g., $C_c^\infty(J)$). A polynomial Stein operator for $W$ is a linear differential operator of the form

$A_W[f](x) = \sum_{j=0}^k p_j(x) f^{(j)}(x)$

with $k<\infty$ and each $p_j(x)$ a real polynomial, such that:

• $A_W[f] \in L^1(W)$ for all $p_W$0,
• $p_W$1 for all $p_W$2.

The set of all such operators is denoted as $p_W$3. For multivariate distributions on $p_W$4, the analogous construction involves polynomials in multiple variables and derivatives along each coordinate.

2. Operator Algebra and Weyl Algebra Structure

Polynomial Stein operators admit a natural embedding into the first Weyl algebra $p_W$5, the noncommutative algebra over $p_W$6 generated by $p_W$7 and $p_W$8 with $p_W$9. Any $J \subseteq \mathbb{R}$0 with $J \subseteq \mathbb{R}$1 is an element of $J \subseteq \mathbb{R}$2 (Azmoodeh et al., 2022). The structure of $J \subseteq \mathbb{R}$3 as a right ideal in $J \subseteq \mathbb{R}$4 allows for rich algebraic and combinatorial analysis, with significant consequences for both explicit operator description and intersection properties.

For the standard normal law, the class $J \subseteq \mathbb{R}$5 is a principal right ideal generated by the classical Gaussian Stein operator $J \subseteq \mathbb{R}$6, i.e., every polynomial Stein operator is $J \subseteq \mathbb{R}$7 for some $J \subseteq \mathbb{R}$8 (Azmoodeh et al., 2022). The real vector space structure is spanned by elements $J \subseteq \mathbb{R}$9, with $\mathcal{F}$0 the Hermite polynomials.

3. Constructing Stein Operators: Algebraic and Algorithmic Methods

For random variables defined as polynomial functions of independent standard normal components, $\mathcal{F}$1 with $\mathcal{F}$2, the existence of polynomial Stein operators connects to the null controllability of a discrete linear control system on the space of polynomials (Azmoodeh et al., 2019). The forward Stein chain recursively constructs candidate coefficient polynomials, and null-control conditions enforce the vanishing expectation property.

An explicit and automatable algorithm computes all algebraic Stein operators up to given operator order $\mathcal{F}$3 and maximal degree $\mathcal{F}$4 in the coefficients:

• Initialize a sequence of candidate polynomials;
• Propagate via algebraic operations consistent with the chain recursion;
• Solve associated block-linear systems to enforce the null-control constraint;
• Extract operator coefficients by polynomial division and moment normalization.

This approach yields explicit high-order Stein operators for polynomial images of Gaussian vectors, e.g., Hermite polynomials $\mathcal{F}$5, including for degrees $\mathcal{F}$6 (Azmoodeh et al., 2019).

4. Combination Rules, Self-adjoint Extensions, and Examples

Polynomial Stein operators exhibit stable algebraic rules under products, powers, and functional compositions of independent random variables. Let $\mathcal{F}$7 and $\mathcal{F}$8. Under the “algebra of Stein operators” (Gaunt et al., 2016):

• If $\mathcal{F}$9 and $C_c^\infty(J)$0 (with $C_c^\infty(J)$1 polynomials in $C_c^\infty(J)$2), then $C_c^\infty(J)$3 characterizes $C_c^\infty(J)$4.
• For sums and iterates, power-increasing and scaling lemmas inductively construct operators with common powers.
• Classical distributions (normal, gamma, beta, variance-gamma) admit first or second order polynomial Stein operators (see Table 1).

Distribution	Stein Operator Form	Operator Order
$C_c^\infty(J)$5	$C_c^\infty(J)$6	1
Gamma $C_c^\infty(J)$7	$C_c^\infty(J)$8	1
Beta $C_c^\infty(J)$9	$W$0	1
Sym. Variance-Gamma	$W$1	2

Higher order self-adjoint differential operators with polynomial coefficients, as constructed by Azad–Laradji–Mustafa (Azad et al., 2014), provide systematic generation of polynomial Stein operators for a wide class of densities $W$2: $W$3 is a Stein operator whenever $W$4 is self-adjoint and $W$5 has density $W$6. This methodology recovers classic Hermite, Laguerre, and Jacobi-type operators and extends naturally to arbitrary even order.

5. Characterizing Properties and Limitations

A polynomial Stein operator $W$7 is said to be characterizing if $W$8 for all test $W$9 implies $A_W[f](x) = \sum_{j=0}^k p_j(x) f^{(j)}(x)$0 has the law of the target (Azmoodeh et al., 2022). For first-order operators with nontrivial polynomial coefficients, the characterizing property generally holds. However, for higher order operators, additional conditions—such as symmetry, infinite divisibility, or moment constraints—may be necessary. For example, all first-order operators of the form $A_W[f](x) = \sum_{j=0}^k p_j(x) f^{(j)}(x)$1 are characterizing for the standard normal, and so are $A_W[f](x) = \sum_{j=0}^k p_j(x) f^{(j)}(x)$2 (Azmoodeh et al., 2022).

A general intersection theorem shows that for any finite collection of random variables with holonomic densities (i.e., densities or characteristic functions solving polynomial-coefficient linear ODEs), their sets of polynomial Stein operators always have nontrivial intersection (Azmoodeh et al., 2022). This implies that there is a Stein operator simultaneously “annihilating” all such distributions—a fact that limits the discriminative power of generic polynomial Stein operators and necessitates further distributional assumptions for characterization.

6. Explicit Stein Operators for Hermite Polynomials and Products

Explicit polynomial Stein operators have been derived for random variables in higher Wiener chaoses, notably for $A_W[f](x) = \sum_{j=0}^k p_j(x) f^{(j)}(x)$3 with $A_W[f](x) = \sum_{j=0}^k p_j(x) f^{(j)}(x)$4 (Gaunt, 2018, Azmoodeh et al., 2019, Azmoodeh et al., 2021). For instance:

• For $A_W[f](x) = \sum_{j=0}^k p_j(x) f^{(j)}(x)$5:

$A_W[f](x) = \sum_{j=0}^k p_j(x) f^{(j)}(x)$6

• For $A_W[f](x) = \sum_{j=0}^k p_j(x) f^{(j)}(x)$7:

$A_W[f](x) = \sum_{j=0}^k p_j(x) f^{(j)}(x)$8

Operators for $A_W[f](x) = \sum_{j=0}^k p_j(x) f^{(j)}(x)$9 with $k<\infty$0 have much higher order, and explicit computation faces rapidly growing technical obstacles.

For products of $k<\infty$1 independent $k<\infty$2 variables, $k<\infty$3, the minimal order polynomial Stein operator is

$k<\infty$4

with $k<\infty$5 the Stirling numbers of the second kind (Azmoodeh et al., 2021).

7. Applications: Discrepancy Measures and Goodness-of-Fit

Polynomial Stein operators underpin the polynomial Stein discrepancy (PSD), a computationally efficient criterion for assessing sample quality and moment matching in Bayesian inference (Srinivasan et al., 2024). Fixing the test space to degree-$k<\infty$6 polynomials $k<\infty$7 and applying the degree-$k<\infty$8 restriction $k<\infty$9 of the canonical Langevin Stein operator yields: $p_j(x)$0 In the Bernstein–von Mises regime, $p_j(x)$1 if and only if $p_j(x)$2 matches the first $p_j(x)$3 moments of $p_j(x)$4. Linear-time computation, empirical power, and tuning-free operation make the PSD a practical alternative to kernel Stein discrepancies for high-dimensional posterior diagnostics and hyperparameter selection (Srinivasan et al., 2024).

• "An algebra of Stein operators" (Gaunt et al., 2016)
• "Polynomial Stein operators: a noncommutative algebra perspective" (Azmoodeh et al., 2022)
• "Stein operators for variables from the third and fourth Wiener chaoses" (Gaunt, 2018)
• "On algebraic Stein operators for Gaussian polynomials" (Azmoodeh et al., 2019)
• "Higher order self-adjoint operators with polynomial coefficients" (Azad et al., 2014)
• "An asymptotic approach to proving sufficiency of Stein characterisations" (Azmoodeh et al., 2021)
• "The Polynomial Stein Discrepancy for Assessing Moment Convergence" (Srinivasan et al., 2024)