Selective Inference

Deep clustering is a powerful approach for discovering meaningful structures in high-dimensional data by learning a low-dimensional latent representation prior to clustering. Despite its empirical success, assessing the statistical reliability of the resulting clusters remains challenging. Testing discovered clusters on the same data induces selection bias and invalidates classical $p$-values. Selective inference (SI) provides a principled framework for correcting this bias, but existing methods focus on clustering performed directly on the observed features. In this work, we develop an SI framework for deep clustering with a fixed pretrained encoder. The key challenge is that cluster assignments are determined through a nonlinear transformation from the original data space to the latent space, resulting in a substantially more complex selection process than in conventional clustering. Our method provides a computationally tractable way to account for this process and enables valid statistical testing of differences between clusters identified in the latent space. Synthetic experiments demonstrate that the proposed method controls the Type I error rate while achieving higher power than valid but conservative baselines, and genomic applications show that it can identify significant cluster differences while appropriately accounting for selection bias. Our framework provides a principled approach to quantifying the statistical reliability of structures discovered by deep clustering.

LLM leaderboard gains can reflect selection among privately evaluated model variants, yet neither the number of variants nor their dependence is public. We ask how many hidden variants a published margin can support while retaining statistical evidence of a provider's advantage over a fixed comparator. For a fixed candidate family under a Gaussian margin model, we derive a sensitivity curve that reports this maximum count as a function of a lower bound on within-family correlation. The relevant correlation must match the score used for ranking and the sampling model: in a controlled family, pooled item correlation is 0.90, whereas composite-score correlation is 0.46 under item resampling and 0.92 when MMLU subjects are resampled. An item-based audit of 394 adjacent-rank claims on the Open LLM Leaderboard finds that 391 lack statistical support even before accounting for selection. Among claims that pass the uncorrected test, certification can depend on assumptions about the hidden family's correlation. The resulting curves make these assumptions explicit without estimating the unobserved search size.

Fully homomorphic encryption (FHE) enables inference on private data without revealing it to the server, but evaluating an entire input under FHE is expensive. We study \emph{selective homomorphic inference}, where only a sensitive region of interest (ROI) is encrypted, and computations independent of that region are performed in plaintext. Selective evaluation produces the same output as full FHE on the same model, without retraining. Its efficiency depends on how quickly encrypted dependencies spread through the network. For small encrypted ROIs, locality-preserving architectures can achieve order-of-magnitude homomorphic-evaluation speedups, whereas architectures with early global mixing provide essentially no speedup. These results identify locality as the key architectural property governing the benefit of selective homomorphic inference.

Selective inference (SI) provides statistically valid $p$-values for hypotheses selected by applying an algorithm to the data, correcting for the bias that arises when the same data are used both to select and to test a hypothesis. Developing an SI procedure for a new algorithm, however, has required an expert to derive, and then implement, the selection event, i.e., the conditions under which the hypothesis is selected. Repeating this specialized effort for every new algorithm is why exact SI has so far been available for only a narrow class. We propose AutoSI, a framework that removes this barrier in two ways. First, AutoSI constructs the selection event automatically from the algorithm's individual operations, so the user only writes the algorithm as ordinary NumPy-like code and derives nothing by hand. Second, AutoSI broadens the class of selection events SI can handle: existing exact methods are limited to selection events characterized by linear or quadratic inequalities in the data, whereas AutoSI covers any algorithm expressible through rational functions of the data (ratios of polynomials). We prove that the $p$-values computed by AutoSI are exactly valid in finite samples. We demonstrate AutoSI on three feature-selection methods, each written in a few dozen lines of code. One of these methods, the lasso with its tuning parameter selected by cross-validated $R^2$, cannot be handled within existing exact SI frameworks and is made possible by AutoSI. Experiments on synthetic and real datasets show that the resulting $p$-values control the type I error rate (i.e., the false positive rate) at the nominal level while retaining high power.

Electroencephalography (EEG) analysis in cognitive studies requires specialized expertise and involves many defensible choices over contrasts, channels, time windows, and statistical tests. LLM agents can translate varied natural-language questions into analysis choices, offering a flexible interface for automation. Yet fluent reports alone cannot establish that an agent selected the requested analysis or evaluated a confirmatory claim independently of adaptive search. We present CogEEGAgent, a cognitive-EEG analysis agent grounded in MNE-Python. Its EEG-specific scientific harness separates semantic from scientific authority. The LLM interprets intent and proposes registered analyses, while deterministic components validate typed contracts, control confirmation access, and authorize evidence-bound release. On a prespecified routing benchmark, CogEEGAgent maps language to registered analyses more accurately than a matched deterministic router, while matched preflight makes both systems abstain whenever required. In an externally model-authored, outcome-blind campaign, the complete system releases supported analyses with participant-disjoint confirmation and blocks prespecified capability hazards and lifecycle-reuse requests. Policy stress testing shows that held-out confirmation curbs false positives from uncorrected adaptive search. Together, these studies establish bounded autonomy and an auditable automation framework for cognitive-EEG workflows. More broadly, they show how scientific agents can combine flexible language understanding with fail-closed control over inference and release.

Knowing how often a language model fails does not explain where its errors concentrate. When auditors examine many explanations, the strongest observed pattern may arise by chance. We introduce Janus, a procedure for checking proposed error patterns before reporting them. Janus starts with a fixed list of yes/no properties of the examples being evaluated, such as whether the input is long. For each property, it compares the model's error rates on examples with that property and those without it. To see how large a difference can arise by chance, it repeats this calculation after shuffling the yes/no labels across examples without changing the group sizes. These shuffled properties are called decoys. A pattern is reported only if the size of its error difference meets a threshold set using decoys. On separate held-out examples, the same group must still have the higher error rate and the difference must meet a minimum, which was chosen in advance. In a controlled experiment, where the model must find a code in documents containing tables of staff, projects, and renewal codes, Janus confirms five related patterns of higher error rates on tasks requiring more lookups across tables. It also confirms a sixth pattern: lower error rates on examples with the needed information at the ends of the tables. In our samples from the MuSiQue and LongBench v2 public benchmarks, SliceLine finds groups with high error rates, while Janus reports no confirmed error patterns for the example properties we chose to test. For comparison, we use standard tests that shuffle errors and account for testing many candidates. With the same holdout check, they confirm two to six controlled patterns, depending on the test and threshold, and none on either benchmark. In simulations with no real error patterns, Janus reports false patterns more often than Benjamini-Hochberg, depending on the decoy count.

Reliably knowing when a language model is correct is almost as important as being correct. We introduce prover-verifier deliberation (PVD), an inference-time protocol grounded in interactive proof theory, as a mechanism for selective prediction: the protocol produces both an answer and a structured confidence verdict, allowing a system to report high-confidence answers while abstaining on uncertain cases. In each dialogue, a prover defends a candidate answer through checkable sub-claims while a verifier issues targeted challenges and returns \textsc{Accept}, \textsc{Challenge}, or \textsc{Reject}. Because frozen language models are imperfect provers and verifiers operating over a noisy channel, formal soundness and completeness guarantees do not transfer; instead, we characterize the protocol empirically through its coverage-precision behavior. Our main experiment uses Claude Sonnet 4.6 as prover and Claude Haiku 4.5 as verifier on GPQA Diamond. Questions accepted with no answer revision, which we call Accept + No Change (ANC), are reported as the high-confidence subset; we evaluate this subset by its precision and coverage. ANC separates reliable from unreliable answers, yielding a $\sim$30pp HC-Prec gap over the non-ANC complement. Robustness experiments with GPT and Gemini pairings show that high HC-Prec can transfer across model families, while verifier strictness and domain competence largely determine the size of the selection gap. On Humanity's Last Exam, weaker prover-verifier pairings can collapse or invert the ANC signal, illustrating a practical failure mode when the verifier operates outside its effective region. Comparisons with self-consistency, universal self-consistency, multi-agent debate, and Reflexion suggest that prover-verifier deliberation supplies a distinct argument-defensibility signal for selective prediction.

The validity of statistical inference depends critically on how data are collected. When data gathered through active data collection (ADC) are reused for a post-hoc inferential task, conventional inference can fail because the sampling is adaptively biased toward regions favored by the collection strategy. This issue is especially pronounced in black-box optimization, where sequential model-based optimization (SMBO) methods such as the tree-structured Parzen estimator (TPE) and Gaussian process upper confidence bound (GP-UCB) preferentially concentrate evaluations in promising regions. We study statistical inference on actively collected data when the inferential target is constructed in a data-dependent manner after data collection. To enable valid inference in this setting, we propose post-ADC inference, a framework that accounts for the biases arising from both the active data collection process and the subsequent data-driven target construction. Our method builds on selective inference and provides valid $p$-values and confidence intervals that correct for both sources of bias. The framework applies to a broad class of ADC processes by imposing only assumptions on the observation noise, without requiring any assumptions on the underlying black-box function or the surrogate model used by the SMBO algorithm. Empirical results also show that post-ADC inference provides valid inference for data collected by GP-UCB and TPE.

Adaptive prompt and program search makes LLM evaluation selection-sensitive. Once benchmark items are reused inside tuning, the observed winner's score need not estimate the fresh-data performance of the full tune-then-deploy procedure. We study inference for this procedure-level target under explicit tuning budgets. We propose SIREN, a selection-aware repeated-split reporting protocol that freezes the post-search shortlist, separates splitwise selection from held-out evaluation, and uses an item-level Gaussian multiplier bootstrap for uncertainty quantification. In a fixed-shortlist regime with smooth stabilized selection, the estimator admits a first-order item-level representation, and the bootstrap yields valid simultaneous inference on a finite budget grid. This supports confidence intervals for procedure-performance curves and pre-specified equal-budget and cross-budget comparisons. Controlled simulations and MMLU-Pro tuning experiments show that winner-based reporting can be optimistic and can change deployment conclusions, while SIREN remains close to the finite-sample reporting target.

Standard selective prediction methods typically estimate uncertainty from the output of a single predictive branch. While effective for general uncertainty estimation, these approaches often struggle under partial observability, where local temporal evidence can be contradictory and standard confidence scores become misleading. We introduce CHASE (Competing Hypotheses for Ambiguity-Aware Selective Prediction), a selective prediction framework that explicitly compares structured temporal explanations to determine whether to commit to a decision or abstain. Because genuine ambiguity causes the score gap between competing hypotheses to collapse, CHASE optimizes a ranking-aware selector over these hypothesis margins to globally separate safe commitments from fundamentally uncertain ones. We evaluate this framework on the problem of hidden connectivity inference, utilizing a controlled, physically grounded simulator inspired by the dynamics of giant unilamellar vesicles (GUVs), alongside zero-shot qualitative transfer (without retraining or fine tuning) to representative real GUV videos. Our experiments demonstrate that explicitly reasoning over competing hypotheses provides a superior balance of metrics. Compared to canonical uncertainty baselines, CHASE achieves statistically significant gains in overall no-abstain accuracy, three-way accuracy, and overall ambiguity-aligned abstention (at 80% coverage). Specifically, it yields up to an 11.0% relative mean improvement in overall alignment, alongside up to an 8.8% relative boost in three-way accuracy in the very-high ambiguity regime. By maintaining a selective risk boundary strictly at par with the best baselines at 80% coverage, and reducing overall risk by 9.9% at 90% coverage, this framework offers a more reliable approach to decision-making under structured ambiguity.

Selective conformal prediction can yield substantially tighter uncertainty sets when we can identify calibration examples that are exchangeable with the test example. In interventional settings, such as perturbation experiments in genomics, exchangeability often holds only within subsets of interventions that leave a target variable "unaffected" (e.g., non-descendants of an intervened node in a causal graph). We study the practical regime where this invariance structure is unknown and must be estimated from data. Our main result quantifies how coverage degrades when the estimated safe calibration set accidentally includes interventions that affect the target, and gives a conservative correction when an upper bound on this error is available. Rather than learning a full causal graph, we learn only the intervention-target relationships needed to choose calibration interventions. We give algorithms for this partial learning task and evaluate them on synthetic structural equation models and Replogle K562 CRISPR-interference data, where the experiments illustrate synthetic gains from selective calibration and finite-sample tradeoffs on real perturbation screens.

Social scientists are increasingly turning to unstructured datasets to unlock new empirical insights, e.g., estimating descriptive statistics of or causal effects on quantitative measures derived from text, audio, or video data. In many settings, unsupervised analysis is of primary interest, in that the researcher does not want to (or cannot) manually pre-specify all important aspects of the unstructured data to measure; they are interested in "discovery." This paper proposes a general and flexible framework for pursuing such discovery from unstructured data in a statistically principled way. The framework leverages recent methods from the literature on AI interpretability to map unstructured data points to high-dimensional, sparse, and interpretable "concept embeddings"; computes statistics from these concept embeddings for testing interpretable, concept-by-concept hypotheses; performs selective inference on these hypotheses using algorithms validated by new results in high-dimensional central limit theory, producing a selected set ("discoveries"); and both generates and evaluates human-interpretable natural language descriptions of these discoveries. The proposed framework has few researcher degrees of freedom, is robust to data snooping and other post-selection inference concerns, and facilitates fast and inexpensive sensitivity analysis and replication. Applications to recent descriptive and causal analyses of unstructured data in empirical economics are explored.