Detecting Nonproperness of Likelihood Equations

Algebraic statistics characterizes statistical models through polynomial constraints, but it has mainly been used for analytically specified model classes. This paper studies the inverse problem: identifying probabilistic structure from vanishing binomials observed in empirical probability tensors. We treat the vanishing binomials of a toric model as its algebraic signature, and turn the ideal-variety correspondence of algebraic statistics into an operational procedure for structural learning that identifies a model by signature matching without parameter estimation. By restricting attention to a computationally tractable class of configuration matrices, which we call {\it the Kronecker-stack class}, we make these signatures explicitly enumerable. Within this class we define minimum invariant constraint (MIC) as the atomic unit characterizing each signature and generalizing the notion of independence. We tested this approach employing MICs on synthetic data as well as on corpus-scale real language data. The results suggested the utility of the method, revealing that the identified rank-one structures correspond to interpretable sets of words. These results open up a new avenue for applying algebraic statistics to computational linguistics.

Empirical investigations into unintended model behavior often show that the algorithm is predicting another outcome than what was intended. These exposés highlight the need to identify when algorithms predict unintended quantities - ideally before deploying them into consequential settings. We propose a falsification framework that provides a principled statistical test for discriminant validity: the requirement that an algorithm predict intended outcomes better than impermissible ones. Drawing on falsification practices from causal inference, econometrics, and psychometrics, our framework compares calibrated prediction losses across outcomes to assess whether the algorithm exhibits discriminant validity with respect to a specified impermissible proxy. In settings where the target outcome is difficult to observe, multiple permissible proxy outcomes may be available; our framework accommodates both this setting and the case with a single permissible proxy. Throughout we use nonparametric hypothesis testing methods that make minimal assumptions on the data-generating process. We illustrate the method in an admissions setting, where the framework establishes discriminant validity with respect to gender but fails to establish discriminant validity with respect to race. This demonstrates how falsification can serve as an early validity check. We also provide analysis in a criminal justice setting, where we highlight the limitations of our framework and emphasize the need for complementary approaches to assess other aspects of construct validity and external validity.

Fractionally supervised classification (FSC) offers a flexible framework for combining labeled and unlabeled data in model-based classification, but existing formulations assume simple random sampling. In many applications, however, the retained observation is an extreme order statistic from a set rather than a randomly selected unit. This is particularly appealing when the target population is rare, since maxima nomination sampling (NS) can enrich the sample with the most informative observations, as in screening, environmental monitoring, repeated testing, and reliability studies. Under such designs, the likelihood function changes fundamentally, and the usual FSC EM construction is no longer valid. We develop FSC for nominated samples by introducing a latent representation that accounts for both the class membership of the observed maximum and the latent composition of the remaining units in the set. The resulting method yields a proper EM algorithm and a coherent weighted-likelihood FSC procedure for NS data. We present the methodology in general form, illustrate it for a rare-event contamination normal mixtures, and show through simulation that it substantially improves on the misspecified alternative by ignoring the extra rank information of such data. A real-data analysis demonstrates its practical value.