Density ratio estimation (DRE) is a paramount task in machine learning, for its broad applications across multiple domains, such as covariate shift adaptation, causal inference, independence tests and beyond. Parametric methods for estimating the density ratio possibly lead to biased results if models are misspecified, while conventional non-parametric methods suffer from the curse of dimensionality when the dimension of data is large. To address these challenges, in this paper, we propose a novel approach for DRE based on the projection pursuit (PP) approximation. The proposed method leverages PP to mitigate the impact of high dimensionality while retaining the model flexibility needed for the accuracy of DRE. We establish the consistency and the convergence rate for the proposed estimator. Experimental results demonstrate that our proposed method outperforms existing alternatives in various applications.
A core motivation of science is to evaluate which scientific model best explains observed data. Bayesian model comparison provides a principled statistical approach to comparing scientific models and has found widespread application within cosmology and astrophysics. Calculating the Bayesian evidence is computationally challenging, especially as we continue to explore increasingly more complex models. The Savage-Dickey density ratio (SDDR) provides a method to calculate the Bayes factor (evidence ratio) between two nested models using only posterior samples from the super model. The SDDR requires the calculation of a normalised marginal distribution over the extra parameters of the super model, which has typically been performed using classical density estimators, such as histograms. Classical density estimators, however, can struggle to scale to high-dimensional settings. We introduce a neural SDDR approach using normalizing flows that can scale to settings where the super model contains a large number of extra parameters. We demonstrate the effectiveness of this neural SDDR methodology applied to both toy and realistic cosmological examples. For a field-level inference setting, we show that Bayes factors computed for a Bayesian hierarchical model (BHM) and simulation-based inference (SBI) approach are consistent, providing further validation that SBI extracts as much cosmological information from the field as the BHM approach. The SDDR estimator with normalizing flows is implemented in the open-source harmonic Python package.
We study the problem of causal function estimation in the Proxy Causal Learning (PCL) framework, where confounders are not observed but proxies for the confounders are available. Two main approaches have been proposed: outcome bridge-based and treatment bridge-based methods. In this work, we propose two kernel-based doubly robust estimators that combine the strengths of both approaches, and naturally handle continuous and high-dimensional variables. Our identification strategy builds on a recent density ratio-free method for treatment bridge-based PCL; furthermore, in contrast to previous approaches, it does not require indicator functions or kernel smoothing over the treatment variable. These properties make it especially well-suited for continuous or high-dimensional treatments. By using kernel mean embeddings, we have closed-form solutions and strong consistency guarantees. Our estimators outperform existing methods on PCL benchmarks, including a prior doubly robust method that requires both kernel smoothing and density ratio estimation.
Density ratio estimation is fundamental to tasks involving $f$-divergences, yet existing methods often fail under significantly different distributions or inadequately overlap supports, suffering from the density-chasm and the support-chasm problems. Additionally, prior approaches yield divergent time scores near boundaries, leading to instability. We design $\textbf{D}^3\textbf{RE}$, a unified framework for robust, stable and efficient density ratio estimation. We propose the dequantified diffusion bridge interpolant (DDBI), which expands support coverage and stabilizes time scores via diffusion bridges and Gaussian dequantization. Building on DDBI, the proposed dequantified Schr{\"o}dinger bridge interpolant (DSBI) incorporates optimal transport to solve the Schr{\"o}dinger bridge problem, enhancing accuracy and efficiency. Our method offers uniform approximation and bounded time scores in theory, and outperforms baselines empirically in mutual information and density estimation tasks.
Density ratio estimation in high dimensions can be reframed as integrating a certain quantity, the time score, over probability paths which interpolate between the two densities. In practice, the time score has to be estimated based on samples from the two densities. However, existing methods for this problem remain computationally expensive and can yield inaccurate estimates. Inspired by recent advances in generative modeling, we introduce a novel framework for time score estimation, based on a conditioning variable. Choosing the conditioning variable judiciously enables a closed-form objective function. We demonstrate that, compared to previous approaches, our approach results in faster learning of the time score and competitive or better estimation accuracies of the density ratio on challenging tasks. Furthermore, we establish theoretical guarantees on the error of the estimated density ratio.
Time-sensitive machine learning benefits from Sequential Probability Ratio Test (SPRT), which provides an optimal stopping time for early classification of time series. However, in finite horizon scenarios, where input lengths are finite, determining the optimal stopping rule becomes computationally intensive due to the need for backward induction, limiting practical applicability. We thus introduce FIRMBOUND, an SPRT-based framework that efficiently estimates the solution to backward induction from training data, bridging the gap between optimal stopping theory and real-world deployment. It employs density ratio estimation and convex function learning to provide statistically consistent estimators for sufficient statistic and conditional expectation, both essential for solving backward induction; consequently, FIRMBOUND minimizes Bayes risk to reach optimality. Additionally, we present a faster alternative using Gaussian process regression, which significantly reduces training time while retaining low deployment overhead, albeit with potential compromise in statistical consistency. Experiments across independent and identically distributed (i.i.d.), non-i.i.d., binary, multiclass, synthetic, and real-world datasets show that FIRMBOUND achieves optimalities in the sense of Bayes risk and speed-accuracy tradeoff. Furthermore, it advances the tradeoff boundary toward optimality when possible and reduces decision-time variance, ensuring reliable decision-making. Code is publicly available at https://github.com/Akinori-F-Ebihara/FIRMBOUND
We propose reinterpreting copula density estimation as a discriminative task. Under this novel estimation scheme, we train a classifier to distinguish samples from the joint density from those of the product of independent marginals, recovering the copula density in the process. We derive equivalences between well-known copula classes and classification problems naturally arising in our interpretation. Furthermore, we show our estimator achieves theoretical guarantees akin to maximum likelihood estimation. By identifying a connection with density ratio estimation, we benefit from the rich literature and models available for such problems. Empirically, we demonstrate the applicability of our approach by estimating copulas of real and high-dimensional datasets, outperforming competing copula estimators in density evaluation as well as sampling.
This paper introduces a novel generalized self-imitation learning ($\textbf{GSIL}$) framework, which effectively and efficiently aligns large language models with offline demonstration data. We develop $\textbf{GSIL}$ by deriving a surrogate objective of imitation learning with density ratio estimates, facilitating the use of self-generated data and optimizing the imitation learning objective with simple classification losses. $\textbf{GSIL}$ eliminates the need for complex adversarial training in standard imitation learning, achieving lightweight and efficient fine-tuning for large language models. In addition, $\textbf{GSIL}$ encompasses a family of offline losses parameterized by a general class of convex functions for density ratio estimation and enables a unified view for alignment with demonstration data. Extensive experiments show that $\textbf{GSIL}$ consistently and significantly outperforms baselines in many challenging benchmarks, such as coding (HuamnEval), mathematical reasoning (GSM8K) and instruction-following benchmark (MT-Bench).
Climate change-driven floods demand advanced forecasting models, yet Graph Neural Networks (GNNs) underutilize river network topology due to tree-like structures causing over-squashing from high node resistance distances. This study identifies this limitation and introduces a reachability-based graph transformation to densify topological connections, reducing resistance distances. Empirical tests show transformed-GNNs outperform EA-LSTM in extreme flood prediction, achieving 24-h water level accuracy equivalent to EA-LSTM's 14-h forecasts - a 71% improvement in long-term predictive horizon. The dense graph retains flow dynamics across hierarchical river branches, enabling GNNs to capture distal node interactions critical for rare flood events. This topological innovation bridges the gap between river network structure and GNN modeling, offering a scalable framework for early warning systems.
Density ratio estimation (DRE) is a fundamental machine learning technique for identifying relationships between two probability distributions. $f$-divergence loss functions, derived from variational representations of $f$-divergence, are commonly employed in DRE to achieve state-of-the-art results. This study presents a novel perspective on DRE using $f$-divergence loss functions by deriving the upper and lower bounds on $L_p$ errors. These bounds apply to any estimator within a class of Lipschitz continuous estimators, irrespective of the specific $f$-divergence loss functions utilized. The bounds are formulated as a product of terms that include the data dimension and the expected value of the density ratio raised to the power of $p$. Notably, the lower bound incorporates an exponential term dependent on the Kullback--Leibler divergence, indicating that the $L_p$ error significantly increases with the Kullback--Leibler divergence for $p > 1$, and this increase becomes more pronounced as $p$ increases. Furthermore, these theoretical findings are substantiated through numerical experiments.
Synthetic data generation is a promising technique to facilitate the use of sensitive data while mitigating the risk of privacy breaches. However, for synthetic data to be useful in downstream analysis tasks, it needs to be of sufficient quality. Various methods have been proposed to measure the utility of synthetic data, but their results are often incomplete or even misleading. In this paper, we propose using density ratio estimation to improve quality evaluation for synthetic data, and thereby the quality of synthesized datasets. We show how this framework relates to and builds on existing measures, yielding global and local utility measures that are informative and easy to interpret. We develop an estimator which requires little to no manual tuning due to automatic selection of a nonparametric density ratio model. Through simulations, we find that density ratio estimation yields more accurate estimates of global utility than established procedures. A real-world data application demonstrates how the density ratio can guide refinements of synthesis models and can be used to improve downstream analyses. We conclude that density ratio estimation is a valuable tool in synthetic data generation workflows and provide these methods in the accessible open source R-package densityratio.
Volumetry is one of the principal downstream applications of 3D medical image segmentation, for example, to detect abnormal tissue growth or for surgery planning. Conformal Prediction is a promising framework for uncertainty quantification, providing calibrated predictive intervals associated with automatic volume measurements. However, this methodology is based on the hypothesis that calibration and test samples are exchangeable, an assumption that is in practice often violated in medical image applications. A weighted formulation of Conformal Prediction can be framed to mitigate this issue, but its empirical investigation in the medical domain is still lacking. A potential reason is that it relies on the estimation of the density ratio between the calibration and test distributions, which is likely to be intractable in scenarios involving high-dimensional data. To circumvent this, we propose an efficient approach for density ratio estimation relying on the compressed latent representations generated by the segmentation model. Our experiments demonstrate the efficiency of our approach to reduce the coverage error in the presence of covariate shifts, in both synthetic and real-world settings. Our implementation is available at https://github.com/benolmbrt/wcp_miccai
Estimating the ratio of two probability densities from a finite number of observations is a central machine learning problem. A common approach is to construct estimators using binary classifiers that distinguish observations from the two densities. However, the accuracy of these estimators depends on the choice of the binary loss function, raising the question of which loss function to choose based on desired error properties. For example, traditional loss functions, such as logistic or boosting loss, prioritize accurate estimation of small density ratio values over large ones, even though the latter are more critical in many applications. In this work, we start with prescribed error measures in a class of Bregman divergences and characterize all loss functions that result in density ratio estimators with small error. Our characterization extends results on composite binary losses from (Reid & Williamson, 2010) and their connection to density ratio estimation as identified by (Menon & Ong, 2016). As a result, we obtain a simple recipe for constructing loss functions with certain properties, such as those that prioritize an accurate estimation of large density ratio values. Our novel loss functions outperform related approaches for resolving parameter choice issues of 11 deep domain adaptation algorithms in average performance across 484 real-world tasks including sensor signals, texts, and images.
The density ratio of two probability distributions is one of the fundamental tools in mathematical and computational statistics and machine learning, and it has a variety of known applications. Therefore, density ratio estimation from finite samples is a very important task, but it is known to be unstable when the distributions are distant from each other. One approach to address this problem is density ratio estimation using incremental mixtures of the two distributions. We geometrically reinterpret existing methods for density ratio estimation based on incremental mixtures. We show that these methods can be regarded as iterating on the Riemannian manifold along a particular curve between the two probability distributions. Making use of the geometry of the manifold, we propose to consider incremental density ratio estimation along generalized geodesics on this manifold. To achieve such a method requires Monte Carlo sampling along geodesics via transformations of the two distributions. We show how to implement an iterative algorithm to sample along these geodesics and show how changing the distances along the geodesic affect the variance and accuracy of the estimation of the density ratio. Our experiments demonstrate that the proposed approach outperforms the existing approaches using incremental mixtures that do not take the geometry of the
We propose a meta-learning method for positive and unlabeled (PU) classification, which improves the performance of binary classifiers obtained from only PU data in unseen target tasks. PU learning is an important problem since PU data naturally arise in real-world applications such as outlier detection and information retrieval. Existing PU learning methods require many PU data, but sufficient data are often unavailable in practice. The proposed method minimizes the test classification risk after the model is adapted to PU data by using related tasks that consist of positive, negative, and unlabeled data. We formulate the adaptation as an estimation problem of the Bayes optimal classifier, which is an optimal classifier to minimize the classification risk. The proposed method embeds each instance into a task-specific space using neural networks. With the embedded PU data, the Bayes optimal classifier is estimated through density-ratio estimation of PU densities, whose solution is obtained as a closed-form solution. The closed-form solution enables us to efficiently and effectively minimize the test classification risk. We empirically show that the proposed method outperforms existing methods with one synthetic and three real-world datasets.
Offline Black-Box Optimization (BBO) aims at optimizing a black-box function using the knowledge from a pre-collected offline dataset of function values and corresponding input designs. However, the high-dimensional and highly-multimodal input design space of black-box function pose inherent challenges for most existing methods that model and operate directly upon input designs. These issues include but are not limited to high sample complexity, which relates to inaccurate approximation of black-box function; and insufficient coverage and exploration of input design modes, which leads to suboptimal proposal of new input designs. In this work, we consider finding a latent space that serves as a compressed yet accurate representation of the design-value joint space, enabling effective latent exploration of high-value input design modes. To this end, we formulate an learnable energy-based latent space, and propose Noise-intensified Telescoping density-Ratio Estimation (NTRE) scheme for variational learning of an accurate latent space model without costly Markov Chain Monte Carlo. The optimization process is then exploration of high-value designs guided by the learned energy-based model in the latent space, formulated as gradient-based sampling from a latent-variable-parameterized inverse model. We show that our particular parameterization encourages expanded exploration around high-value design modes, motivated by inversion thinking of a fundamental result of conditional covariance matrix typically used for variance reduction. We observe that our method, backed by an accurately learned informative latent space and an expanding-exploration model design, yields significant improvements over strong previous methods on both synthetic and real world datasets such as the design-bench suite.
Estimating the ratio of two probability densities from finitely many samples, is a central task in machine learning and statistics. In this work, we show that a large class of kernel methods for density ratio estimation suffers from error saturation, which prevents algorithms from achieving fast error convergence rates on highly regular learning problems. To resolve saturation, we introduce iterated regularization in density ratio estimation to achieve fast error rates. Our methods outperform its non-iteratively regularized versions on benchmarks for density ratio estimation as well as on large-scale evaluations for importance-weighted ensembling of deep unsupervised domain adaptation models.
A prominent family of methods for learning data distributions relies on density ratio estimation (DRE), where a model is trained to $\textit{classify}$ between data samples and samples from some reference distribution. DRE-based models can directly output the likelihood for any given input, a highly desired property that is lacking in most generative techniques. Nevertheless, to date, DRE methods have failed in accurately capturing the distributions of complex high-dimensional data, like images, and have thus been drawing reduced research attention in recent years. In this work we present $\textit{classification diffusion models}$ (CDMs), a DRE-based generative method that adopts the formalism of denoising diffusion models (DDMs) while making use of a classifier that predicts the level of noise added to a clean signal. Our method is based on an analytical connection that we derive between the MSE-optimal denoiser for removing white Gaussian noise and the cross-entropy-optimal classifier for predicting the noise level. Our method is the first DRE-based technique that can successfully generate images beyond the MNIST dataset. Furthermore, it can output the likelihood of any input in a single forward pass, achieving state-of-the-art negative log likelihood (NLL) among methods with this property. Code is available on the project's webpage in https://shaharYadin.github.io/CDM/ .
Density ratio estimation (DRE) is a fundamental machine learning technique for capturing relationships between two probability distributions. State-of-the-art DRE methods estimate the density ratio using neural networks trained with loss functions derived from variational representations of $f$-divergences. However, existing methods face optimization challenges, such as overfitting due to lower-unbounded loss functions, biased mini-batch gradients, vanishing training loss gradients, and high sample requirements for Kullback--Leibler (KL) divergence loss functions. To address these issues, we focus on $\alpha$-divergence, which provides a suitable variational representation of $f$-divergence. Subsequently, a novel loss function for DRE, the $\alpha$-divergence loss function ($\alpha$-Div), is derived. $\alpha$-Div is concise but offers stable and effective optimization for DRE. The boundedness of $\alpha$-divergence provides the potential for successful DRE with data exhibiting high KL-divergence. Our numerical experiments demonstrate the effectiveness of $\alpha$-Div in optimization. However, the experiments also show that the proposed loss function offers no significant advantage over the KL-divergence loss function in terms of RMSE for DRE. This indicates that the accuracy of DRE is primarily determined by the amount of KL-divergence in the data and is less dependent on $\alpha$-divergence.
Consider a scenario where we have access to train data with both covariates and outcomes while test data only contains covariates. In this scenario, our primary aim is to predict the missing outcomes of the test data. With this objective in mind, we train parametric regression models under a covariate shift, where covariate distributions are different between the train and test data. For this problem, existing studies have proposed covariate shift adaptation via importance weighting using the density ratio. This approach averages the train data losses, each weighted by an estimated ratio of the covariate densities between the train and test data, to approximate the test-data risk. Although it allows us to obtain a test-data risk minimizer, its performance heavily relies on the accuracy of the density ratio estimation. Moreover, even if the density ratio can be consistently estimated, the estimation errors of the density ratio also yield bias in the estimators of the regression model's parameters of interest. To mitigate these challenges, we introduce a doubly robust estimator for covariate shift adaptation via importance weighting, which incorporates an additional estimator for the regression function. Leveraging double machine learning techniques, our estimator reduces the bias arising from the density ratio estimation errors. We demonstrate the asymptotic distribution of the regression parameter estimator. Notably, our estimator remains consistent if either the density ratio estimator or the regression function is consistent, showcasing its robustness against potential errors in density ratio estimation. Finally, we confirm the soundness of our proposed method via simulation studies.