Mutual Information (MI) is a fundamental metric for quantifying dependency between two random variables.

Successful applications ofInfoNCE (Information Noise-ContrastiveEstimation) and its variants have popularized the use of contrastive variational mutual information (MI) estimators in machine learning.

We propose a fully data-driven approach to designing mutual information (MI) estimators. Since any MI estimator is a function of the observed sample from two random variables, we parameterize this function with a neural network (MIST) and train it end-to-end to predict MI values. Training is performed on a large meta-dataset of 625,000 synthetic joint distributions with known ground-truth MI. To handle variable sample sizes and dimensions, we employ a two-dimensional attention scheme ensuring permutation invariance across input samples. To quantify uncertainty, we optimize a quantile regression loss, enabling the estimator to approximate the sampling distribution of MI rather than return a single point estimate. This research program departs from prior work by taking a fully empirical route, trading universal theoretical guarantees for flexibility and efficiency. Empirically, the learned estimators largely outperform classical baselines across sample sizes and dimensions, including on joint distributions unseen during training. The resulting quantile-based intervals are well-calibrated and more reliable than bootstrap-based confidence intervals, while inference is orders of magnitude faster than existing neural baselines. Beyond immediate empirical gains, this framework yields trainable, fully differentiable estimators that can be embedded into larger learning pipelines. Moreover, exploiting MI's invariance to invertible transformations, meta-datasets can be adapted to arbitrary data modalities via normalizing flows, enabling flexible training for diverse target meta-distributions.

Mutual information (MI) is one of the most general ways to measure relationships between random variables, but estimating this quantity for complex systems is challenging. Denoising diffusion models have recently set a new bar for density estimation, so it is natural to consider whether these methods could also be used to improve MI estimation. Using the recently introduced information-theoretic formulation of denoising diffusion models, we show the diffusion models can be used in a straightforward way to estimate MI. In particular, the MI corresponds to half the gap in the Minimum Mean Square Error (MMSE) between conditional and unconditional diffusion, integrated over all Signal-to-Noise-Ratios (SNRs) in the noising process. Our approach not only passes self-consistency tests but also outperforms traditional and score-based diffusion MI estimators. Furthermore, our method leverages adaptive importance sampling to achieve scalable MI estimation, while maintaining strong performance even when the MI is high.

The InfoNCE objective, originally introduced for contrastive representation learning, has become a popular choice for mutual information (MI) estimation, despite its indirect connection to MI. In this paper, we demonstrate why InfoNCE should not be regarded as a valid MI estimator, and we introduce a simple modification, which we refer to as InfoNCE-anchor, for accurate MI estimation. Our modification introduces an auxiliary anchor class, enabling consistent density ratio estimation and yielding a plug-in MI estimator with significantly reduced bias. Beyond this, we generalize our framework using proper scoring rules, which recover InfoNCE-anchor as a special case when the log score is employed. This formulation unifies a broad spectrum of contrastive objectives, including NCE, InfoNCE, and $f$-divergence variants, under a single principled framework. Empirically, we find that InfoNCE-anchor with the log score achieves the most accurate MI estimates; however, in self-supervised representation learning experiments, we find that the anchor does not improve the downstream task performance. These findings corroborate that contrastive representation learning benefits not from accurate MI estimation per se, but from the learning of structured density ratios.

Estimating mutual information (MI) is a fundamental task in data science and machine learning. Existing estimators mainly rely on either highly flexible models (e.g., neural networks), which require large amounts of data, or overly simplified models (e.g., Gaussian copula), which fail to capture complex distributions. Drawing upon recent vector copula theory, we propose a principled interpolation between these two extremes to achieve a better trade-off between complexity and capacity. Experiments on state-of-the-art synthetic benchmarks and real-world data with diverse modalities demonstrate the advantages of the proposed estimator.

However, estimating MI between multidimensional variables is challenging because the number of samples necessary to converge to an accurate estimate scales unfavorably with dimensionality.