Conditional Independence

Conditional independence tests (CITs) test for conditional dependence between two random objects $X$ and $Y$ given a third random object $Z$. Existing CITs have limited applicability to high-dimensional data, especially multimodal data like text. However, we show that such tests are of interest for large language model (LLM) outputs, where we test whether an output $X$ generated from a source text $Z$ carries information about an attribute $Y$ beyond $Z$ itself. For this purpose, we propose embedded CITs (eCITs), which embed $X$ and $Z$ and apply an existing CIT to the resulting representations and to $Y$. We show that, provided the embedding of $Z$ is sufficient, i.e. retains the information $Z$ carries about either $Y$ or the representation of $X$, the null hypothesis transfers from $X$ and $Z$ to their representations, so that a CIT valid for the embedded hypothesis is valid for the original one. We further give conditions for equivalence of the two hypotheses, and show that sufficiency weakens to mean sufficiency when the embedded test targets conditional mean independence. We propose a semi-synthetic simulation design to assess type I error (T1E) control and power of the eCITs for given embedding maps on a specific dataset and task, and use it to evaluate them on our application. Applying the eCITs to German Parliament speeches, we find for all combinations of embedding maps considered that the summaries of two LLMs contain information about the speaker's faction and gender beyond the speech they were generated from.

An increasing number of scholars use AI to measure variables they subsequently include in downstream analyses. Although AI-measured variables are often analyzed as if observed without error, ignoring prediction errors in automated measurement leads to substantial bias and invalid confidence intervals in downstream analyses, even if AI measurement accuracy is high, e.g., above 90%. Existing solutions, such as design-based supervised learning and prediction-powered inference, combine error-prone AI-based measurements with gold-standard labels, which may be costly and difficult to obtain in some application areas. In this paper, we propose debiased inference with multiple imperfect measurements (DMM), a framework that combines multiple error-prone AI measurements to enable valid downstream inference without gold-standard labels. Building on the established results on CP decomposition, DMM assumes that these measurements are independent conditional on the latent true label and observed unit-level features, such as text features represented by embeddings. This framework allows for unknown misclassification rates to vary across annotation methods (e.g., large language models) and across units of annotation (e.g., texts). Under this assumption, we use semiparametric inference theory to prove that the DMM estimator is consistent and asymptotically normal, enabling valid inference for a wide range of downstream statistical analyses common in the social sciences. Our simulation results show that DMM yields valid inference and that adding accurate, though imperfect, measurements can improve efficiency. Focusing on common applications of large language model annotations, we also develop diagnostics to assess the conditional independence assumption.

We present the mathematical foundations of linear independent component analysis (ICA) models based on standard literature in a self-contained note. It is aimed at readers with a background in measure-theoretic probability theory. We first develop the theory of the characteristic functions of probability measures on $\mathbb{R}^d$, including their analyticity and the way in which they determine and characterise the distributions. We then focus on several identifiability results of ICA models with successively strengthened assumptions on the sources: from merely non-constant, to non-Gaussian, to Gaussian-free independent sources. Under the strictest assumptions, we show that the independent sources are identifiable up to translation, permutation, scales and signs, and this even in the presence of additive Gaussian noise. Furthermore, we present the online equivariant gradient descent ICA algorithm for recovering the independent sources from data, in the standard complete noiseless non-Gaussian ICA setting.

Compositional reliability bounds for multi-agent systems multiply component reliabilities, a step licensed by a conditional-independence assumption that is routinely stated and rarely tested. We test it. Two instances of one model, in a two-agent handoff, co-fail on 90.0% of the missions on which either fails (log OR 6.66, 95% CI [6.38, 7.00]; phi 0.916), in a preregistered evaluation of 18,000 missions scored by deterministic code with no LLM judge. Substituting a different model reduces the association in six of six contrasts; substituting a different vendor, model already different, does not -- a registered hypothesis reported as a null. The error is signed and runs against the operator: positive dependence inflates joint failure above the independence product, so redundancy is over-credited exactly when components share a model. The assumption-free alternative is often vacuous, and fitting a dependence model is worse: we prove a bootstrap bound on a fitted model's functional loses coverage of the truth as n grows, the identification gap being O(1) while the bootstrap haircut is O(n^{-1/2}). More data makes such a certificate worse, with no visible symptom. We give a finite-sample certificate assuming no dependence structure: a linear program over the joint, over a Bonferroni-Clopper-Pearson box around measured co-execution moments. It is sound, sharp for the information supplied, and monotone in the moment family. Enriching ten moment functionals to fourteen narrows the identified interval by 85.7% and lifts the certified floor from 0.2455 to 0.4116. A companion anytime-valid certificate holds type-I error at 0.0471 under optional stopping. Common dependence statistics are marginal-bounded and can reverse an apparent ordering of conditions when the compared agents fail at different rates. Contracts, scoring code, analysis scripts, and the preregistration are released.

What a finite learning device has recorded and what will hold value for it on future tasks are not the same quantity. We develop a typed accounting for finite-state learning devices that separates four components: a training-side fit functional $Φ_{\mathrm{fit}}$, the record-correlation stock $J_{D}=I(M;D)$, an update-side search ledger $σ_{M}$, and an operational capital value $V(M;T,b)$. This value is the work gap between an informed protocol class and a blind class obtained by deleting the memory-read port and re-optimizing from scratch. (I) Separation: for every $n$, there is a device family on which record correlation and world correlation grow by $n\ln 2$ while the capital gain is exactly zero. In the $\mathrm{flat}^{*}$ regime, data-free updates never increase $V$. (II) Capitalization ledger: an exact $\mathrm{flat}^{*}$ extraction identity and a universal ledger identity give, for (F5$'$)-stable $M$-local updates under a no-discarded-record-correlation condition (f), the bound $η_{\mathrm{cap}}\le 1$ for the capitalization efficiency $η_{\mathrm{cap}}=ΔV/(k T\,σ_{M})$, together with necessary and sufficient conditions for equality. (III) Value retention: for the retention gap $L_{\mathrm{gen}}$ and retention ratio $ρ_{\mathrm{gen}}$ (the former carries no sign constraint; the latter is defined for positive training-side value and is not confined to $[0,1]$) we give a two-layer alignment domain: an exact exchange rate between value and the side-information-adjusted record fit $I(M';D\mid Y)$ without any record-side-information independence assumption, and a raw record-stock exchange rate under a joint side-information neutrality condition $(M,D)\perp Y$, whose boundary is marked by an explicit one-time-pad witness. These are statements about finite-device value retention under task-distribution shift, not a theory of statistical generalization.

Conditional Independence (CI) tests are the statistical engine of constraint-based causal discovery: in algorithms such as PC (Peter-Clark) and FCI (Fast Causal Inference), skeleton pruning and key orientations follow directly from CI decisions. This survey reviews CI testing with emphasis on assumptions, robustness, and scalability in high-dimensional and mixed-type settings common in biomedical domains. The survey organizes widely used CI methods into six families: partial-correlation, contingency-table, regression, nearest-neighbor, kernel, and machine-learning-based. Special emphasis is provided on the robustness layers that address the limitations of these families. For each family, the survey examines when CI decisions reflect the data-generating distribution and when they fail. By this, we link test-level properties, including power decay with conditioning set size and asymmetric type I/II error consequences, to graph-level errors in skeleton recovery and v-structure orientation. The survey also compares adoption across major R and Python libraries and summarizes open challenges, including mixed-type CI testing without discretization, small-sample error control, and strategies for improving scalability of CI-testing.

The problem of learning the graphical Markov blanket (MB) of a variable from data has applications in many areas such as structure learning for Bayesian networks and Markov random fields, causal discovery, and feature selection. However, a common assumption most methods make is that the conditional independencies in the distribution imply the same separation in the graphical structure -- also known as the faithfulness assumption. Unfortunately, this assumption can be violated by higher-order dependencies such as XOR and parity-type relations, and -- on finite samples -- by empirical violations that, in extreme cases, even induce spurious dependencies absent from the true distribution. Therefore, in this paper we propose a "k-order" relaxation of the faithfulness assumption that captures parity type relationships between k+2 variables. We then propose a proof of concept algorithm called k-order Markov blanket (kOMB) that uses this relaxation for MB discovery. Finally, we empirically show how kOMB can recover the MB of a variable under both true and empirical violations of faithfulness. Code available at: https://github.com/lklee9/k-order-Markov-blanket

Graphical models are ubiquitous in social and empirical science as they are intuitive and easy to use. These models belong to the broader class of Markov models, defined using solely conditional independence (CI) restrictions. In order to estimate finite-dimensional target parameters in such models efficiently, semi-parametric theory provides a principled framework for constructing regular and asymptotically linear estimators via influence functions (IFs). These estimators are asymptotically normal and root-$n$ consistent. Characterizing the class of all influence functions for a target parameter is crucial for statistically efficient inference in these models. For models that are Markov relative to directed acyclic graphs (DAGs), the orthogonal complement of the tangent space is known, implying that for any target the class of all influence functions can be derived once an influence function is obtained. On the other hand, for Markov models not equivalent to a DAG model -- such as ordinary Markov models associated with undirected graphs, chain graphs, or acyclic directed mixed graphs -- the orthogonal complement has not been characterized, impeding semi-parametric inference in these models. We derive closed form expressions for the orthogonal complement of the tangent space for general Markov models and illustrate our results by characterizing the class of influence functions for the conditional mean parameter in several graphical models.

We propose CEDAR (Causal Edge Discovery for Autoregressive Processes), a constraint-based method for lagged causal edge discovery in sparse autoregressive time series. CEDAR screens candidate cross-variable lags using AR(1)-residualized, U-centered distance correlation, then applies two targeted conditional-independence tests per significant cross-variable lag candidate and accepts at most one lag per ordered pair. A stable MCI pruning step removes indirect edges, and optional deterministic C-nodes adjust for specified trend-like nonstationarity. In sparse regimes where few lags survive screening, CEDAR requires $O(d^2)$ CI tests after screening while retaining edge-level interpretability. CEDAR is most effective when data are scarce and variables exhibit lag-1 self-dynamics; methods with richer conditioning sets become preferable as $T$ grows or when higher-order autoregressive or simultaneous multi-lag effects are common.

This paper proposes a causal independence principle for value -- the value Causal Markov Condition (v-CMC) -- and develops the conceptual and mathematical foundations of a "causal value theory" linking causality and utility. After motivating a local formulation of the v-CMC, we introduce a probability-value duality that translates standard causal-inference results into the value setting. In particular, we formulate local, global, and decomposition versions of the v-CMC and prove their equivalence. We also define v-separation and show that it is sound and complete for conditional value independence. Furthermore, we derive a Bellman-type recursion as a special case of the v-CMC, thereby generalizing standard Bellman recursion from linear chains to causal DAGs. Finally, we show how the v-CMC supports modular transfer and updating of utility information across causal contexts and develop algorithms for causally structured utility elicitation and canonical influence-diagram construction.

Causal discovery algorithms learn a network that describes the causal dependencies among random variables. A common workflow involves first utilizing conditional independence properties on observational data to determine partially directed causal relationships, then applying interventions to orient the unknown causal directions. A critical assumption for the first step is faithfulness: a requirement that causally linked variables exhibit statistical dependence. Many natural systems include buffering and stabilizing pathways that cancel out to achieve systemic robustness. This cancellation of pathways violates faithfulness, leading causal discovery algorithms to incorrectly remove causal dependencies. In this paper, we argue that hard interventions contain information about the presence/absence of causal linkage that is overlooked in the first stage of structure discovery. We show that a mild assumption -- called intervention-immediacy faithfulness -- that allows cancellations, is sufficient to nonparametrically identify causal structures with hard interventions. These results position interventions as the primary carriers of information about causal structure, which should take precedence over conditional independence testing. To flip the paradigm, we also specify equivalence classes when the identification criteria are not met due to limitations in the scope of interventions.

We study a broad class of graphical models whose independencies correspond to vertex separation in mixed graphs with directed, undirected, and bidirected edges, that are capable of encoding independence structures arising from feedback, latent and selection mechanisms. In particular, we introduce separable graphs, in which each missing edge implies the existence of a separating set for its endpoints, and essentially separable graphs, those graphs separation equivalent to a separable graph. We show that these models include many existing graph families used to define graphical models an provide several characterizations of separable graphs and essentially separable graphs. We also provide multiple characterizations of separation equivalence for separable graphs. One is a graphical characterization in terms of ordinary graph properties, extending earlier results for specific subfamilies Another is a separational characterization depending only on graph separation properties. Finally, we provide a canonical representation for the equivalence classes of essentially separable graphs and develop an algorithm that, under suitable assumptions, identifies the equivalence class of any essentially separable graph.

Model cards quote trust-benchmark scores without recording when they were measured, and the same number is routinely carried across successive checkpoints of one release line as if the model behind it had not shifted. We test whether it has shifted by auditing four open-source release lines, Yi, Qwen, Mistral, and Gemma, at three successive generations each, on a fixed basket of trust benchmarks under multiple prompt templates. Mean absolute adjacent-generation drift lands well above an independence-based no-drift reference null, and the gap persists when we drop a benchmark, drop a release line, or switch to strict scoring. We therefore conclude that a trust score attached to a release line should not be carried forward to the next checkpoint without remeasurement; it should instead be reported as a checkpoint-bound, dated artefact, which we package as a longitudinal model card. Closed APIs, larger models, canonical benchmark protocols, and fixed month-cadence rules lie outside the audited scope and require their own evaluation.

We study the task of learning the structure of a $d$-sparse Gaussian graphical model on $n$ variables from a single trajectory of Glauber dynamics. Beyond algorithmic considerations, many applications present temporally correlated observations rather than i.i.d.\ samples. In the classical i.i.d.\ setting, under comparably general sparsity and minimum edge-strength assumptions, sublinear-in-$n$ sample guarantees are known, but achieving them in polynomial-time remains open. Motivated in part by this gap, we give a polynomial-time algorithm that recovers the conditional-independence graph from a single Glauber trajectory, with a trajectory-length guarantee that does not depend on the mixing time. Technically, our algorithm has three components. First, we estimate the conditional variances and rescale the trajectory to reduce to the unit-diagonal case, without changing the underlying graph. Second, we design a local edge test that extracts adjacency information from short update windows by isolating pairwise influence. Third, we aggregate these local statistics using a robust median-based estimator, and prove accuracy despite temporal dependence arising from a single trajectory.

Statistical matching combines partially overlapping datasets that share covariates $X$ but observe the target $Y$ and auxiliary variables $Z$ separately. Classical approaches typically invoke the conditional independence assumption (CIA), which makes the problem identifiable but fundamentally implies that the imported auxiliary variable provides no additional predictive power for $Y$ once $X$ is known. To capture this latent $Y$--$Z$ dependence, we propose a novel dependency-aware Schrödinger bridge for predictive statistical matching. Our approach couples the two separated databases by tilting the conservative CIA baseline with a transportation-based compatibility cost, recovering an informative joint distribution. The resulting statistical learning framework yields full probabilistic posterior rules for bidirectional imputation. Theoretically, we establish a sufficient condition under which the learned bridge strictly improves over the CIA baseline, alongside an exact joint recovery guarantee in the Gaussian setting under an appropriate cost. Across synthetic benchmarks and real-world datasets (CelebA and Adult), we demonstrate that our dependency-aware completion consistently improves downstream predictive utility, proving especially beneficial in settings like data recoding where the underlying population exhibits strong $Y$--$Z$ dependence.

Testing conditional independence is fundamental yet intrinsically difficult: without additional assumptions, Type I error control is impossible in general. The "Model-X'' paradigm addresses this difficulty by assuming exact knowledge of a relevant conditional distribution. While small deviations from this assumption can sometimes be tolerated in classical one-shot testing, existing sequential conditional independence tests typically require the Model-X conditional to be known exactly, making them fragile when it must instead be estimated. We propose a new approach that is substantially more robust to such estimation error. Our method applies testing-by-betting to an adaptively optimized Kernel Conditional Independence statistic, together with a normalization scheme and a truncate-and-shift calibration strategy. These modifications greatly reduce Type I error inflation while preserving high power across high-dimensional synthetic benchmarks and real-world fairness tasks, outperforming existing sequential Model-X approaches. Code is available at https://github.com/he-zh/SKCI.

Constraint-based causal discovery relies on repeated conditional independence tests, but fast nonparametric tests often sacrifice calibration, especially when variables depend on the conditioning set through nonlinear relationships. We introduce BLITZ (Broad-to-Local Independence Testing via residualiZation), a nonparametric conditional independence test designed to run well under a second while maintaining the accuracy needed for the thousands of queries performed by constraint-based causal discovery algorithms. BLITZ first removes broad smooth dependence on the conditioning set using low-order polynomial regression, then applies a small nonlinear feature map and residualizes those features with shallow tree regressions. The resulting statistic tests residual cross-covariance, with a moment-matched chi-square approximation to the null distribution. We show theoretically that the two-stage design reduces the effective complexity faced by the tree residualizers, allowing shallow trees to control residual conditional-mean bias while avoiding excessive overfitting. In simulations, BLITZ provides better null calibration than fast kernel, random-feature, and regression-based competitors while remaining among the fastest methods tested. In causal discovery experiments on synthetic graphs and flow-cytometry data, BLITZ yields more reliable endpoint orientations among retained adjacencies and competitive structural recovery. These results suggest that broad-to-local residualization is a practical route to calibrated, scalable nonparametric conditional independence testing for causal discovery.

We study component recovery and mixing-matrix estimation from unlabeled finite mixtures whose observable distributions share the same latent components but have unknown mixing weights. The main identifying signal is marginal independence: each component is assumed to be independent on at least one coordinate pair, but no labels, clean component samples, or mixing weights are observed. We first prove a structural result for product components: under linear independence of the univariate marginals, any independent affine combination of the components must coincide with a single component. We then extend this principle to observable mixtures and show that, under full-rank and no-cancellation conditions, marginally independent affine combinations recover the corresponding latent components. When every component is independent on some coordinate pair, all components are identifiable, and the mixing matrix is recoverable under the stated completion conditions. Finally, we propose a Product-Marginal Maximum Mean Discrepancy (PM-MMD) estimator over affine combinations of the observable mixtures and prove uniform convergence and stability under approximate marginal independence. This framework also separates the empirical roles of the assumptions: irreducibility is, in general, not directly testable from the unlabeled mixtures alone, whereas marginal independence yields a candidate-level diagnostic through held-out PM-MMD. Controlled and flow-cytometry experiments show when marginal independence provides a useful recovery signal. In the reported multi-component comparisons, condition-aware representative selection stabilizes PM-MMD and improves recovery relative to clustering, factorization, and pairwise mixture-proportion baselines using the same unlabeled mixtures.

Exploiting the indistinguishability of objects in a probabilistic graphical model such as a factor graph is key to lifted probabilistic inference algorithms and allows for tractable probabilistic inference problems with respect to domain sizes. A central building block for the exploitation of indistinguishable objects in factor graphs is the identification of commutative factors, i.e., factors whose output values are invariant under permutations of input values assigned to a subset of their arguments. In this paper, we revisit the theoretical foundations underlying the state-of-the-art algorithm to detect commutative factors. Specifically, we show that in its current form, the state-of-the-art algorithm relies on a central theorem that is mistakenly regarded as a sufficient condition to identify commutative factors, while it actually only implies necessary condition. Consequently, the state of the art might, as we show in this paper, deliver incorrect results. To fix the flaws currently present in the state of the art, we prove a slightly modified version of the aforementioned theorem, which serves as a necessary condition to identify commutative factors. Moreover, we present a corrected version of the state-of-the-art algorithm, which keeps its efficiency while ensuring correctness and introduce a complementary algorithm with tighter worst-case bounds.

The Hilbert-Schmidt Independence Criterion (HSIC) and its joint-independence extension $d\mathrm{HSIC}$ are degenerate $V$-statistics whose data-dependent weighted-$χ^2$ null limits force a permutation calibration that multiplies the per-test cost by the number of permutations, in practice two orders of magnitude. Adapting the recent martingale MMD construction for two-sample testing to the (joint) independence problem, we introduce two studentised statistics whose null distributions are standard normal regardless of the data law, so that a single normal-quantile lookup replaces the permutation step entirely. The first, $m\mathrm{HSIC}$, is a self-normalised lower-triangular sum of the Hadamard product of two empirically centred Gram matrices. Under independence and bounded-fourth-moment kernels it converges to a standard normal. It is consistent against every fixed alternative, and runs at quadratic cost in the sample size without any sample split, matching the biased HSIC $V$-statistic. Our second statistic, $md\mathrm{HSIC}$, achieves finite-sample consistency with a single half-sample split: the centring is estimated on one half and the lower-triangular self-normalised martingale is run on the other, shrinking the conditional-mean residual to a quantity that is exponentially small in $d$, so the statistic is asymptotically standard normal at every fixed number of jointly tested variables, with a per-test cost that grows only linearly in $d$. On synthetic data with per-variable input dimension from $1$ to $500$ and between $2$ and $10$ jointly tested variables, both statistics match the empirical type-I error rate and test power of permutation-calibrated baselines while running $25$ to $60\times$ faster.

Identifying latent variables and the causal structure involving them is essential across various scientific fields. While many existing works fall under the category of constraint-based methods (with e.g. conditional independence or rank deficiency tests), they may face empirical challenges such as testing-order dependency, error propagation, and choosing an appropriate significance level. These issues can potentially be mitigated by properly designed score-based methods, such as Greedy Equivalence Search (GES) (Chickering, 2002) in the specific setting without latent variables. Yet, formulating score-based methods with latent variables is highly challenging. In this work, we develop score-based methods that are capable of identifying causal structures containing causally-related latent variables with identifiability guarantees. Specifically, we show that a properly formulated scoring function can achieve score equivalence and consistency for structure learning of latent variable causal models. We further provide a characterization of the degrees of freedom for the marginal over the observed variables under multiple structural assumptions considered in the literature, and accordingly develop both exact and continuous score-based methods. This offers a unified view of several existing constraint-based methods with different structural assumptions. Experimental results validate the effectiveness of the proposed methods.

Synthetic persona datasets cite alignment with official demographics as a basis for trust, yet downstream users consume them as joint structures across age, sex, region, occupation, education, name, and institutional status. Marginal alignment does not imply that these joints are preserved. We propose the Independence-Assumption Footprint (IAF), an audit primitive that operates on the attribute combinations a dataset card itself documents as treated independently. For each such combination, IAF compares the synthetic joint against an external official or institutional reference, using direct joint tables where available and rule-implied checks otherwise. Applied to NVIDIA Nemotron-Personas-Korea (one million Korean synthetic personas), IAF finds that NPK aligns with KOSIS marginals while three joints fail. The major-by-occupation distribution against the KEIS graduate universe carries a large conditional mismatch. The age profile of military service is institutionally inconsistent. Female representation in male-dominated occupations is substantially over-flattened toward parity, with the strict screening verdict mapping-dependent and age-robust under direct standardisation. A transferability demonstration across six further NPK locales finds locale-dependent rather than universal diagnostics, with reference-taxonomy cardinality confounding cross-locale flag counts. For synthetic personas used as silicon samples, marginal claims must therefore be paired with disclosure-anchored joint audits before reuse. The released audit artefacts (reference manifests, occupational crosswalks, derived metrics, reproducibility scripts) instantiate this protocol on the NPK family and are released for retargeting at other synthetic persona resources.

We consider the problem of testing properties of graphs underlying high-dimensional graphical models. We adopt the model of covariance queries introduced by Lugosi, Truszkowski, Velona, and Zwiernik (2021). We study the case when the underlying graph is a tree. The main results of the paper show that, while reconstructing the entire tree may be costly, certain global structural properties can be tested efficiently. In particular, we design randomized tests for global structural properties that use a sub-quadratic number of queries. We develop testing procedures for several fundamental properties, including the number of leaves, the maximum degree, the typical distance, and the diameter of the tree. For each property, we obtain explicit query complexity bounds that depend on the target threshold and tolerance parameters.

Constraint-based causal discovery is widely used for learning causal structures, but heavy reliance on conditional independence (CI) testing makes it computationally expensive in high-dimensional settings. To mitigate this limitation, many divide-and-conquer frameworks have been proposed, but most assume causal sufficiency, i.e., no latent variables. In this paper, we show that divide-and-conquer strategies can be theoretically generalized beyond causal sufficiency to settings with latent variables. Specifically, we propose a recursive decomposition framework, termed DiCoLa, that enables divide-and-conquer causal discovery in the presence of latent variables. It recursively decomposes the global learning task into smaller subproblems and integrates their solutions through a principled reconstruction step to recover the global structure. We theoretically establish the soundness and completeness of the proposed framework. Extensive experiments on synthetic data demonstrate that our approach significantly improves computational efficiency across a range of causal discovery algorithms, while experiments on a real-world dataset further illustrate its practical effectiveness.

Estimating causal effects from observational data requires identifying valid adjustment sets. This task is especially challenging in realistic settings where latent confounding and feedback loops are present. Existing approaches typically assume acyclicity or rely on global causal structure learning, limiting applicability and computational efficiency. In this work, we study a local, data-driven method for covariate selection based on conditional independence information. While this method is known to be sound and complete in acyclic causal models, its validity in the presence of cycles has remained unclear. Our main contribution is to show that these guarantees extend to cyclic causal models. In particular, our result relies on the invariance of conditional independence assertions under $σ$-acyclification. These findings establish a unified, cycle-agnostic perspective on covariate selection and causal effect estimation, showing that the method applies across cyclic and acyclic settings without modification. Empirically, we validate this on extensive synthetic data, showing reliable performance in cyclic causal models.

Finite-sample analyses of deep Q-learning typically treat replayed data as independent, even though it is sampled from temporally dependent state-action trajectories. We study the Deep Q-networks (DQN) algorithm under explicit dependence by modelling the minibatches used for updating the network as $τ$-mixing. We show that this assumption holds under certain dependence conditions on the underlying trajectories and the mechanism used to sample minibatches. Building on this observation, we extend statistical analyses of DQN with fully connected ReLU architectures to dependent data. We formulate each update as a nonparametric regression problem with $τ$-mixing observations and derive finite-sample risk bounds under this dependence structure. Our results show that temporal dependence leads to a degradation in the statistical rate by inducing an additional dimensionality penalty in the rate exponent, reflecting the reduced effective sample size of $τ$-mixing data. Moreover, we derive the sample complexity of DQN under $tau$-mixing from these risk bounds. Finally, we empirically demonstrate on standard Gymnasium environments that the independence assumption is systematically violated and that replay sampling yields approximately exponentially decaying correlations, supporting our theoretical framework.

Gaussian process (GP) marginal likelihood scores and kernel conditional independence tests are theoretically appealing for nonlinear causal discovery but computationally prohibitive at scale. We present three complementary RFF-based methods forming a practical toolkit for score-based, constraint-based, and hybrid causal discovery. The Fourier Feature Marginal Likelihood (FFML) score approximates the exact GP marginal likelihood by replacing the $n x n$ kernel Gram matrix with a finite-dimensional feature representation, reducing cost to $O(nm^2 + m^3)$ while retaining the probabilistic interpretation and automatic complexity penalty of the exact score. FFML extends to mixed (continuous and discrete) parent sets via a product-kernel construction, with a Kronecker path for small discrete parent sets and a Hadamard-product path otherwise. The Tetrad Random Fourier Feature (TRFF) score is a complementary BIC-style alternative using penalized Student-t regression with random Fourier features. TRFF offers robustness to heavy-tailed noise and faster runtime than FFML. Empirically, TRFF and FFML exhibit a complementary precision-recall profile: TRFF achieves higher precision while FFML achieves better recall and lower SHD overall. The Fourier Feature Conditional Independence (FFCI) test is a fast nonparametric CI test for mixed data, using ridge residualization in feature space and a Frobenius-norm cross-covariance statistic approximated as a weighted sum of chi-squared variables. Empirically, BOSS+FFML achieves the lowest SHD on nonlinear data, while BOSS+TRFF offers the highest precision. When run through PC-Max, FFCI and RCIT exhibit complementary precision-recall profiles: RCIT is more precise while FFCI achieves better recall and substantially lower SHD, at approximately twice the runtime.

The standard constraint-based paradigm for causal discovery with incomplete data -- impute first, test second -- is frequently miscalibrated: any consistent conditional independence (CI) test rejects a true null with probability approaching 1 when imputation error induces spurious conditional dependence. We introduce PAIR-CI, a nonparametric CI test that restores calibration by integrating multiple imputation directly into the inferential procedure via a paired permutation design. PAIR-CI compares cross-validated models that include and exclude the candidate variable while receiving the same imputed conditioning set, forcing imputation error to cancel in their loss difference rather than contaminate the test statistic. A provably consistent variance estimator jointly accounts for uncertainty arising from cross-validation and multiple imputation -- to our knowledge, the first formal unification of these two inferential frameworks. In simulations, existing imputation-based CI tests exhibit false positive rates of 28--45% when data are missing not at random (MNAR), whereas PAIR-CI averages below the nominal 5% level across data-generating processes and missingness mechanisms. These gains are largest in nonlinear settings and grow with causal graph size: when integrated into the PC algorithm, PAIR-CI reduces structural Hamming distance by 8% on 10-variable nonlinear graphs, 15% on 30-variable equivalents, and up to 44% on the 56-variable HAILFINDER network, with stable performance in all settings.

Causal discovery methods such as LiNGAM identify causal structure from observational data by assuming mutually independent disturbances. This assumption is fragile: shared volatility, common scale effects, or other forms of dependence can cause the methods to recover the wrong causal order, even with infinite data. We introduce the Linear Mean-Independent Acyclic Model (LiMIAM), which replaces full independence with weaker one-sided mean-independence restrictions on the disturbances. Under finite-order consequences of these restrictions, source nodes are generically identifiable, and hence a compatible causal order can be recovered recursively. Our proof is constructive and leads to DirectLiMIAM, a sequential residual-based algorithm for causal discovery under dependent noise. In simulations with mean-independent but dependent disturbances, DirectLiMIAM outperforms LiNGAM methods. A large-scale empirical application to the oil market highlights the implausibility of the independence assumption and the ability of DirectLiMIAM to recover a realistic causal ordering, from policy to production and from prices to inflation.

Deep learning methods have proved highly effective for classification and image recognition problems. In this paper, we ask whether this success can be transferred to hypothesis testing: if a neural network can distinguish, for example, an image of a handwritten digit from another, can it also distinguish an "image of a sample" (such as a scatter plot) generated under a given statistical model from one generated outside that model? Motivated by this idea, we propose a novel procedure called deep-testing, which approaches the classical inferential problem of hypothesis testing through deep learning. More specifically, the test statistic is a classification map learned by a deep neural network from simulated data satisfying the null and alternative hypotheses, leveraging its strong discriminating power to construct a highly powerful test. As a proof of concept, we apply deep-testing to the problem of independence testing, arguably one of the most important problems in statistics. In a large-scale simulation study, deep-testing achieves the highest overall power against nineteen competing methods across a broad range of complex dependence structures, confirming the viability of the proposed approach.

Constraint-based causal discovery is brittle in finite-sample regimes because erroneous conditional-independence (CI) decisions can cascade into substantial structural errors. We propose Quantitative Argumentation for Causal Discovery (QACD), a semantics-driven framework that represents CI outcomes as graded, defeasible arguments rather than irreversible constraints. QACD maps statistical test outcomes to argument strengths and aggregates conflicting evidence through connectivity-mediated witness propagation, producing a fixed-point acceptability labeling over candidate adjacencies. Experiments on standard benchmark Bayesian networks suggest that QACD improves structural coherence and interventional reliability in several noisy or inconsistent CI regimes, while remaining competitive with classical constraint-based, hybrid, and prior argumentation-based baselines.

Conditional independence testing is a critical component of feature screening, invariant statistical models, and causal discovery. Many of these algorithms rely on the sequential application of conditional independence tests, and their stability hinges on how their outcomes interact. We study this meta-dependence'' between conditional independence properties using the following geometric intuition: satisfying each conditional independence property constrains the space of possible joint distributions to a manifold. The meta-dependence'' of multiple conditional independences in a probability distribution is informed by its position relative to these manifolds. We provide a simple-to-compute measure of this meta-dependence using moment projections, with a closed-form expression for multivariate Gaussian distributions, and consolidate our findings empirically using both synthetic and real-world data. Our measure of meta-dependence does not rely on graphical properties of the distribution and can be computed directly from summary statistics such as a covariance matrix, allowing for various applications. We demonstrate one use case of meta-dependence, using a simple redundancy metric to tune significance thresholds and improve causal discovery.

Deep learning models exhibit fairness concerns when predictions are inadvertently influenced by sensitive attributes. However, existing attempts to make Path-Specific Counterfactual Fairness optimizable rely on estimating marginal potential outcome probabilities-an approach that fundamentally requires high-dimensional conditional density estimation and breaks down in modalities such as medical images, where the curse of dimensionality renders reliable estimation infeasible. To address this limitation, we reduce the problem of enforcing Path-Specific Counterfactual Fairness to a causal conditional independence constraint and prove that satisfying this constraint is sufficient to eliminate the unfair causal effect. This reduction replaces intractable counterfactual estimation with a discriminative optimization objective that remains scalable in high-dimensional settings.