In the upper-level, the fair predictor is updated to beclose toallsubgroup specific predictors. Wefurther provethat such abilevel objective can effectively control the group sufficiency and generalization error.

However, because they are not identifiable without any assumptions, various assumptions have been utilized to evaluate the joint probabilities of potential outcomes, e.g., the assumption of monotonicity (Pearl, 2009; Tian and Pearl, 2000), the independence between potential outcomes (Robins and Richardson, 2011), the condition of gain equality (Li and Pearl, 2019), and the specific functional relationshipsbetween cause and effect (Pearl, 2009). Unlike existing identification conditions, in order to evaluate the joint probabilities of potential outcomeswithoutsuch assumptions,this paper proposestwo types of novel identification conditions using covariate information. In addition, when the joint probabilities of potential outcomes are identifiable through the proposed conditions, the estimation problem of the joint probabilities of potential outcomes reduces to that of singular models and thus they can not be evaluated by standard statistical estimation methods. To solve the problem,this paper proposes a new statisticalestimationmethod based on the augmented Lagrangianmethod and shows the asymptoticnormality of the proposed estimators. Given space constraints, the proofs, the details on the statistical estimationmethod, some numerical experiments, and the case study are provided in the supplementary material.

This might tell us, for instance, how typically we could expect failure of a method that depended on these assumptions.

Multilingual language models achieve strong aggregate performance yet often behave unpredictably across languages, scripts, and cultures. We argue that mechanistic explanations for such models should satisfy a \emph{causal} standard: claims must survive causal interventions and must \emph{cross-reference} across environments that perturb surface form while preserving meaning. We formalize \emph{reference families} as predicate-preserving variants and introduce \emph{triangulation}, an acceptance rule requiring necessity (ablating the circuit degrades the target behavior), sufficiency (patching activations transfers the behavior), and invariance (both effects remain directionally stable and of sufficient magnitude across the reference family). To supply candidate subgraphs, we adopt automatic circuit discovery and \emph{accept or reject} those candidates by triangulation. We ground triangulation in causal abstraction by casting it as an approximate transformation score over a distribution of interchange interventions, connect it to the pragmatic interpretability agenda, and present a comparative experimental protocol across multiple model families, language pairs, and tasks. Triangulation provides a falsifiable standard for mechanistic claims that filters spurious circuits passing single-environment tests but failing cross-lingual invariance.

Modern machine learning embeddings provide powerful compression of high-dimensional data, yet they typically destroy the geometric structure required for classical likelihood-based statistical inference. This paper develops a rigorous theory of likelihood-preserving embeddings: learned representations that can replace raw data in likelihood-based workflows -- hypothesis testing, confidence interval construction, model selection -- without altering inferential conclusions. We introduce the Likelihood-Ratio Distortion metric $Δ_n$, which measures the maximum error in log-likelihood ratios induced by an embedding. Our main theoretical contribution is the Hinge Theorem, which establishes that controlling $Δ_n$ is necessary and sufficient for preserving inference. Specifically, if the distortion satisfies $Δ_n = o_p(1)$, then (i) all likelihood-ratio based tests and Bayes factors are asymptotically preserved, and (ii) surrogate maximum likelihood estimators are asymptotically equivalent to full-data MLEs. We prove an impossibility result showing that universal likelihood preservation requires essentially invertible embeddings, motivating the need for model-class-specific guarantees. We then provide a constructive framework using neural networks as approximate sufficient statistics, deriving explicit bounds connecting training loss to inferential guarantees. Experiments on Gaussian and Cauchy distributions validate the sharp phase transition predicted by exponential family theory, and applications to distributed clinical inference demonstrate practical utility.