Mutual Information Estimation

Mutual information (MI) provides an objective for suppressing or encouraging statistical dependence in implicit generative models. However, direct MI evaluation is challenging in implicit models due to typically intractable densities. A remedy is estimating the generator gradient from the difference between conditional and marginal scores. This score difference can, in turn, be estimated by differentiating a log density ratio learned through classification. This construction nevertheless faces two difficulties: (i) singular distributions need not admit the required score functions, and (ii) poor overlap can hinder density-ratio estimation. We therefore introduce Spread Mutual Information (SMI), a weighted integral of MI across noise levels obtained by applying a common spreading kernel to the generated variable. Gaussian spreading yields smooth, strictly positive conditional and marginal densities, extending the gradient construction to distributions that may originally be singular. Across a variaty of experiments, SMI consistently achieves effective dependence control among MI-based methods and remains competitive with established task-specific approaches.

Pointwise Mutual Information (PMI) is a core measure of testing word association, and Skip-gram with Negative Sampling (SGNS) is essentially a method that implicitly factorizes a shifted PMI matrix. However, a systematic and well-rounded characterization of finite-sample uncertainty in PMI estimation remains absent and imperative to venture into. We provide a statistical framework for PMI estimation and its connection to SGNS. We prove consistency, asymptotic unbiasedness, and asymptotic normality of the empirical PMI estimator, derive its variance via the Delta method, and, applying stochastic approximation theory, obtain a variance decomposition for SGNS-based PMI estimation that separates data variance from optimization variance. Simulation experiments validate the Delta method approximation. Real-data experiments on the Brown Corpus (d = 100) reveal that SGNS systematically deviates from the theoretical relationship PMI + log K. The empirical relationship shows an attenuated PMI coefficient, an amplified log K effect, and a positive intercept, indicating systematic bias. Word analogy validation confirms the models are effective. The failure to validate the variance decomposition under low-dimensional conditions does not diminish its theoretical value; rather, it identifies the unbiasedness assumption as the key bottleneck and clarifies the gap between asymptotic theory and practice, providing implications for both practice and theory.

Estimating mutual information (MI) from samples is a central objective in a variety of scientific fields. Modern neural estimators are accurate in the large-data regime, but they fall short when data is scarce, and each must be fit anew for every distribution under study. Current estimators are moreover tied to specific data types. These constraints limit their adoption in many applications where per-distribution training is impractical and sample sizes are small. We present ALICE, a foundation model that removes per-distribution training, while achieving competitive estimation accuracy. Trained exclusively on a broad family of synthetic distributions, ALICE acts as an in-context estimator of rectified-flow velocity fields: conditioned on samples of an unseen distribution, it estimates that distribution's velocity field without any explicit training. MI is then obtained through a fixed identity that integrates the squared difference between the joint and conditional fields. We validate ALICE on a standard, challenging benchmark and apply it in three domains, biology, genetics, and neuroscience, whose data the model has never seen. For the first time, we show that a single model closes the gap with neural estimators trained separately for each distribution, while natively supporting different data dimensionality and sample cardinality, enabling zero-shot MI analysis across scientific domains.

Multimodal forecasting models that combine time series with text annotations promise richer prediction through textual context, but how do we know whether a text annotation meaningfully contributes to the forecasters prediction? This is an information-theoretic question, but to evaluate whether information-theoretic metrics can reliably measure the predictive value an annotation provides, a ground truth benchmark is needed, and none currently exist. We create a synthetic time series signal with annotations in three categories: semantically correct, incorrect, and irrelevant. Because the data generation process is fully controlled, ground-truth information content is known exactly, enabling principled evaluation of six complementary mutual information estimators (KSG, MINE, InfoNCE, CCA, PID and V-information). We show that all six estimators identify correct annotations as most informative, and are able to audit the quality of mixed text corpora, choosing the annotations that result in the best downstream forecasting results without the need for model training. Our benchmark identifies limitations of each estimator, and these are validated on seven real-world datasets, which show how estimator performance differs on weak signals. Finally, we establish practical rules for implementing these metrics for annotation auditing and fusion selection.

Mutual information is a general measure of statistical dependence that captures both linear and nonlinear relationships between random variables. For continuous and multidimensional variables For continuous multidimensional variables, mutual information must be estimated from samples. Because mutual information is unbounded, its values are not directly comparable across datasets, dimensions, or applications. Normalized mutual information addresses this limitation by converting mutual information into a normalized dependency score. Recent work has demonstrated the practical value of normalized mutual information in applications such as molecular dynamics {arXiv:2405.04980} and interpretable machine learning {arXiv:2409.16768}, but existing estimators remain sensitive to dimensionality and numerical stability {arXiv:2410.07642}. In this paper, we propose a fully neural normalized mutual information estimator for continuous variables. The proposed approach combines a MINE-based neural mutual information estimator {arXiv:1801.04062} with MI-NEE-inspired neural marginal entropy estimators {arXiv:1905.12957}. Mutual information is estimated using the Donsker--Varadhan representation, while marginal entropies are estimated by learning the divergence between each marginal distribution and a uniform reference distribution, from which entropy is recovered. The resulting estimator provides a neural alternative to k-nearest-neighbor-based normalized mutual information estimation {arXiv:2405.04980}. Experiments on Gaussian data from one to eight dimensions show that the proposed estimator improves accuracy over a KSG-based normalized mutual information baseline. These results indicate that neural estimation is a promising direction for normalized dependency measurement in continuous multidimensional settings.

Mutual information (MI) estimation is a central problem in machine learning and statistics; however, existing benchmarks typically evaluate estimators on simplified, low-dimensional distributions, leaving their performance on complex, realistic data largely unexplored. We address this gap with a comprehensive benchmarking framework grounded in a unified copula-theoretic perspective that subsumes existing benchmarks as special cases. Within this framework, we propose two complementary families of tests: a copula-first family that systematically varies ground-truth MI, dimensionality, and marginal complexity using synthetic and flow-based transformations; and a marginals-first family that couples real-world image data with controlled dependency structures, extending the classic same-class-pairing paradigm. We use this suite to extensively evaluate three classes of estimators: non-parametric, discriminative, and generative. Contrary to prevailing assumptions, our results indicate that there is no universal winner: each category can systematically outperform all other estimators under specific setups. By analyzing these cases, we identify fundamental estimation barriers and propose new tests that more effectively stress these specific limitations. We share the open source code at https://github.com/VanessB/mutinfo.

Uncovering the true informational architecture of real-world complex systems requires disentangling how their components uniquely store, redundantly share, and synergistically integrate information over time. Integrated Information Decomposition ($Φ$ID) is a framework for decomposing the information dynamics of multivariate systems into sixteen non-overlapping atoms that characterize redundant, unique, and synergistic modes of information storage, transfer, and integration. Existing methods to compute $Φ$ID are restricted to Gaussian or discrete systems, preventing its application to continuous non-Gaussian dynamical systems. We address this limitation by proposing DIPHINE (Diffusion-based $Φ$-ID Neural Estimator), the first neural estimator that leverages score-based diffusion models to jointly estimate all the mutual information terms required by $Φ$ID from a single amortized network, recovering the sixteen atoms through Möbius inversion. We provide a theoretical analysis of error propagation through the inversion, showing that the Jacobian of the mapping from mutual informations to atoms is integer-valued and that the synergy-to-synergy atom is provably the hardest to estimate. We demonstrate accurate recovery of ground-truth atoms on synthetic benchmarks, superior performance compared to established mutual information estimators, and the ability to extract physiologically interpretable information-dynamic structure on an application involving real data without any distributional assumptions.

Pairwise dependence measures such as correlation and causality are fundamental to temporal data mining, yet there is still no principled and robust way to quantify dependence between heterogeneous data types, especially between continuous time series and discrete temporal event sequences. Existing approaches rely on ad hoc transformations or mutual-information estimators that are highly sensitive to quantization, repeated values, and event redundancy, leading to biased or unstable results in practice. We propose a nonparametric mutual information estimator that directly measures the dependence between time series and event sequences without data transformation, learning, or ad hoc discretization. Our method models the continuous-discrete duality of real-world time series to handle quantization and repeated-value artifacts and introduces a latent event clustering strategy to mitigate bias from event co-occurrence and redundancy. Together, these yield a robust and unified framework that bridges discrete and continuous mutual information. We evaluate the proposed estimator on four representative tasks: discrete-continuous time-delayed mutual information for causality analysis, global and local temporal repetition discovery, discrete covariate selection for time series forecasting, and continuous feature selection for classification. Experiments on synthetic and real-world datasets show consistent improvements over existing methods in accuracy, robustness, and interpretability, positioning our approach as a general-purpose dependence operator for heterogeneous temporal data, similar to Pearson correlation for homogeneous time series. Code available at: https://github.com/HaojiHu/Multimodal-Temporal-Data-Quantification

Measuring statistical dependency between high-dimensional random variables is a fundamental task in data science and machine learning. Neural mutual information (MI) estimators offer a promising avenue, but they typically require costly iterative optimization for each new dataset, making them impractical for real-time applications. We present InfoAtlas, a foundation model-like architecture that eliminates this bottleneck by directly inferring MI in a single forward pass. Pretrained on large-scale synthetic data with rich dependence patterns, InfoAtlas learns to identify diverse dependence structures and predict MI directly from the dataset. Comprehensive experiments demonstrate that InfoAtlas matches state-of-the-art neural estimators in accuracy while achieving $100\times$ speedup, can flexibly handle varying dimensions and sample sizes through a single unified model, and generalizes effectively to complex, real-world scenarios. By reformulating MI estimation as an inference task, InfoAtlas establishes a foundation for real-time dependency analysis.

Estimating mutual information from text usually requires training a task-specific critic, which limits its use in low-data settings. We ask whether large language models can instead estimate pointwise mutual information zero-shot, using only prompts and elicited probabilities. We construct a benchmark from three publicly available human-annotated datasets with ground-truth PMI, and evaluate five information-theoretic prompting-based estimators. Our main method, PromptNCE, frames conditional probability estimation as a contrastive task and augments the candidate set with an explicit OTHER category. The OTHER category allows the model to assign probability mass outside the candidate set, avoiding the closed-set normalization of standard contrastive prompts. PromptNCE gives the best conditional probability estimates on all three datasets. For full PMI, we find that estimating label base rates is the primary bottleneck on two of the three datasets, with the best methods reaching Spearman correlation up to 0.78. We also present a case study in computer science education showing how these estimators can be used to score student knowledge summaries in a low-data setting. We release our code and prompts.

Active learning for continuous regression has lacked an acquisition function that targets epistemic uncertainty when the predictive distribution is multimodal: variance misses modal disagreement, and information-theoretic targets like BALD are designed for discrete outputs. We introduce a Two-Index framework that makes this separation explicit: one stochastic index selects among competing model hypotheses (epistemic source), while a second governs within-hypothesis randomness (aleatoric source). An entropy decomposition within the framework identifies the mutual information between the output and the epistemic index as a principled acquisition objective, and we prove this quantity vanishes as the model is trained on growing datasets, confirming that it captures exactly the uncertainty data can resolve. Because this mutual information is intractable for continuous outputs, we derive the Mutual Information Lower Bound (MI-LB) acquisition function, a closed-form approximation for Mixture Density Network ensembles. On benchmarks featuring multimodal systems, MI-LB matches or beats every baseline evaluated and is the only method to do so consistently -- geometric and Fisher-based baselines compete only when the input space already encodes the multimodality, and collapse otherwise.

Resource-efficient machine learning increasingly uses sparse Mixture-of-Experts (MoE) architectures, where the gate acts as both a learning component and a routing interface controlling computation, communication, and accuracy. Motivated by finite-rate interpretations of MoE gating, we treat the gate as a stochastic channel and use $I(X;T)$ to quantify the routing information available to the selected expert. To make the associated information quantities tractable beyond synthetic examples, we develop a finite-bank MNIST construction using pretrained CNN experts and a discrete, data-dependent selection rule. Since the selected model belongs to a finite candidate set, the algorithmic mutual information $I(S;W)$ admits a closed-form discrete-entropy estimator from the empirical posterior $q(W|S)$. Sweeping a data-dependence parameter $α$, we observe that $\widehat I(S;W)$ monotonically tracks the generalization gap, while the Xu-Raginsky bound exhibits the expected looseness. We also compare with a uniform union-bound baseline and introduce an empirical estimator of $I(X;T)$ together with a Blahut-Arimoto procedure for tracing an accuracy-rate curve over the expert bank. The proposed framework provides a practical tool for analyzing resource-aware MoE inference systems and for interpreting $I(X;T)$ and $D(R_g)$ as design proxies for efficient expert routing.

Information plane (IP) analysis has been suggested to study the training dynamics of deep neural networks through mutual information (MI) between inputs, representations, and targets. However, its statistical validity is often compromised by the difficulty of estimating MI from samples of high-dimensional, deterministic representations. In this work, we perform IP analyses on binary neural networks (BNNs) where activations are discrete and MI is finite. We characterise the finite-sample behaviour of the plug-in entropy estimator and identify regimes for sample size $N$ and representation dimensionality $D$ under which MI estimates are reliable. Outside these regimes, we show that empirical MI estimates saturate to $\log_2 N$, rendering IP trajectories uninformative. Restricting attention to the reliable regime, we train 375 BNNs to investigate the existence of late-stage compression phases and the relationship between compressed representations and generalisation performance. Our results show that while late-stage compression is frequently observed, compressed latent representations do not consistently correlate with improved generalization performance. Instead, the relationship between compression and generalisation is highly dependent on task, architecture, and regularisation.

Reporting guidance for information-theoretic measures is rarely tested against ground truth. We test one guardrail in two multi-agent reinforcement learning games, a social dilemma and a coordination race, where directed influence between selected agent pairs is zero by construction, over 100 seeds. Omitting one precondition, exclusion of the non-stationary training transient, gives false-positive rates of 100.00% and 99.95%: agents annealing exploration independently, in runs that never met, are flagged as influencing one another. Excluding the transient reaches 3.0% in the social dilemma but 11.8% in the coordination game, which stationarity tests explain: 95.7% of social-dilemma series are stationary afterwards against 56.8% of coordination series. So the non-stationarity must be treated, and exclusion is neither the only way nor sufficient. What we recommend instead changes the null model rather than the data: permuting the source within blocks of training time reaches 5.25% and 5.50%, the only one of four constructions at the size of the test in both games, leaving series, statistic and estimand untouched. Titrating injected links of known strength in both games shows it is also the most sensitive of the three, detecting 89.0% in the coordination game where conditioning detects 61.0% on identical pairs, while the ablated test reports 100% with or without a link, so its apparent sensitivity is uninformative. The block count is not critical: every setting from 16 to 256 lands in the nominal region, and a partition derived from the stationarity test removes the parameter, though less sensitively. Code and data are released.

In safety-critical classification, the cost of failure is often asymmetric, yet Bayesian deep learning summarises epistemic uncertainty with a single scalar, mutual information (MI), that cannot distinguish whether a model's ignorance involves a benign or safety-critical class. We decompose MI into a per-class vector $C_k(x)=σ_k^{2}/(2μ_k)$, with $μ_k{=}\mathbb{E}[p_k]$ and $σ_k^2{=}\mathrm{Var}[p_k]$ across posterior samples. The decomposition follows from a second-order Taylor expansion of the entropy; the $1/μ_k$ weighting corrects boundary suppression and makes $C_k$ comparable across rare and common classes. By construction $\sum_k C_k \approx \mathrm{MI}$, and a companion skewness diagnostic flags inputs where the approximation degrades. After characterising the axiomatic properties of $C_k$, we validate it on three tasks: (i) selective prediction for diabetic retinopathy, where critical-class $C_k$ reduces selective risk by 34.7% over MI and 56.2% over variance baselines; (ii) out-of-distribution detection on clinical and image benchmarks, where $\sum_k C_k$ achieves the highest AUROC and the per-class view exposes asymmetric shifts invisible to MI; and (iii) a controlled label-noise study in which $\sum_k C_k$ shows less sensitivity to injected aleatoric noise than MI under end-to-end Bayesian training, while both metrics degrade under transfer learning. Across all tasks, the quality of the posterior approximation shapes uncertainty at least as strongly as the choice of metric, suggesting that how uncertainty is propagated through the network matters as much as how it is measured.

Mutual information (MI) is a principled and widely used objective for robotic information gathering (RIG), providing strong theoretical guarantees for sensor placement (SP) and informative path planning (IPP). However, its high computational cost - dominated by repeated log-determinant evaluations - has limited its use in real-time planning. This paper presents Schur-MI, a Gaussian process (GP) MI formulation that (i) leverages the iterative structure of RIG to precompute and reuse expensive intermediate quantities across planning steps, and (ii) uses a Schur-complement factorization to avoid large determinant computations. Together, these methods reduce the per-evaluation cost of MI from $\mathcal{O}(|\mathcal{V}|^3)$ to $\mathcal{O}(|\mathcal{A}|^3)$, where $\mathcal{V}$ and $\mathcal{A}$ denote the candidate and selected sensing locations, respectively. Experiments on real-world bathymetry datasets show that Schur-MI achieves up to a $12.7\times$ speedup over the standard MI formulation. Field trials with an autonomous surface vehicle (ASV) performing adaptive IPP further demonstrate the method's practicality. By making MI computation tractable for online planning, Schur-MI helps bridge the gap between information-theoretic objectives and real-time robotic exploration. Our code is available at: www.sgp-tools.com

We evaluate artificial intelligence (AI) systems without ground truth by exploiting a link between strategic gaming and information loss. Building on established information theory, we analyze which mechanisms resist adversarial manipulation. This motivates mutual evaluation, where the overseer is treated as a strategic player estimating mutual information by prompting, making truthful agent reporting an optimal strategy. We show that certain f-divergences, such as total variation distance (TVD), maintain polynomial guarantees under attack, building on an established exponential barrier for estimating mutual information (MI) in worst-case certification settings. Under adversarial attacks, TVD-MI maintains effectiveness (area under the curve 0.70--0.77) while other approaches can decay toward chance, demonstrating that prompting the same system for information relationships rather than quality judgments can improve robustness. The mechanisms decompose pairwise evaluations into reliable item-level detection scores without ground truth, addressing a key limitation of standard peer prediction. Pre-registration: https://osf.io/c7pum .