Generalized method of moments
Concrete Problems in AI Safety — inherited
A prior econometric estimation framework (Hansen 1982) that identifies model parameters from moment conditions without fully specifying the underlying distribution.
Hansen's generalized method of moments, from 1982, lets an econometrician pin down a model's parameters using only a handful of equations the true parameters must satisfy, no full description of the data's distribution required. It reaches this wiki as one of the partially specified approaches Concrete Problems in AI Safety (2016) surveys.
Identification from moment conditions alone occupies the first section; the second shows how the method generalizes the paper's founding example.
Identifying parameters from moment conditions alone
The generalized method of moments (GMM), developed by Lars Peter Hansen in 1982, is an econometric estimation framework that identifies a model's parameters from a set of moment conditions — equations of the form $\mathbb{E}[g(x; \theta_0)] = 0$ that the true parameter $\theta_0$ must satisfy — without requiring the analyst to specify the full probability distribution generating the data. Given a moment function $g$ and data $\{x_i\}_{i=1}^n$, the GMM estimator minimizes a quadratic form in the sample moments, $$\hat{\theta} = \arg\min_{\theta} \left(\frac{1}{n}\sum_{i=1}^n g(x_i;\theta)\right)^{\!\top} W \left(\frac{1}{n}\sum_{i=1}^n g(x_i;\theta)\right),$$ for some positive semi-definite weighting matrix $W$; Hansen showed this estimator is consistent and asymptotically normal under general conditions, and that an optimal choice of $W$ minimizes its asymptotic variance. GMM predates Concrete Problems by over three decades and belongs to a large family of econometric tools for partial specification that also includes limited-information maximum likelihood and instrumental-variables estimation (concrete-problems, §"Partially specified models: method of moments, unsupervised risk estimation, causal identification, and limited-information maximum likelihood.", p. 18).
Generalizing the paper's founding example
Concrete Problems introduces Partially specified models through a single worked example: a linear regression $y = \langle w^*, x \rangle + v$ where only $\mathbb{E}[v \mid x] = 0$ is assumed about the noise $v$, with no distributional form specified — already enough to identify $w^*$ under distributional shift (concrete-problems, §"Partially specified models: method of moments, unsupervised risk estimation, causal identification, and limited-information maximum likelihood.", p. 17). The edge Generalized method of moments — generalizes the founding example of → Partially specified models identifies this single moment condition as exactly the $g(x;\theta) = 0$ template GMM was built to handle at scale, combining many such conditions and supplying the general machinery, consistency and identification proofs included, that lets partial specification move beyond toy cases. A second edge, Generalized method of moments — diverges in target from → Unsupervised risk estimation, contrasts it with Unsupervised risk estimation: GMM answers "what is the true model," while unsupervised risk estimation answers the different question "how wrong is this model," a divergence that shows how flexible the umbrella idea of partial specification really is. The concept is a child of Partially specified models within the Weakening statistical assumptions to survive distribution shift theme, sits in both the Borrowed machinery that doesn't fully transfer, and says so and One umbrella idea, aimed at different targets each time it's used connective themes, and belongs to the Distributional shift: relax the statistics, or rewrite the contract supertheme as the historical and mathematical anchor of the paper's partial-specification family.