We propose Partition Tree, a novel tree-based framework for conditional density estimation over general outcome spaces that supports both continuous and categorical variables within a unified formulation. Our approach models conditional distributions as piecewise-constant densities on data-adaptive partitions and learns trees by directly minimizing conditional negative log-likelihood. This yields a scalable, nonparametric alternative to existing probabilistic trees that does not make parametric assumptions about the target distribution. We further introduce Partition Forest, a bagging extension obtained by averaging conditional densities. Empirically, we demonstrate improved probabilistic prediction over CART-style trees and competitive performance compared to state-of-the-art probabilistic tree methods and Random Forests.
Personalized medicine seeks to identify the causal effect of treatment for a particular patient as opposed to a clinical population at large. Most investigators estimate such personalized treatment effects by regressing the outcome of a randomized clinical trial (RCT) on patient covariates. The realized value of the outcome may however lie far from the conditional expectation. We therefore introduce a method called Dirac Delta Regression (DDR) that estimates the entire conditional density from RCT data in order to visualize the probabilities across all possible outcome values. DDR transforms t…
In categorical data analysis, several regression models have been proposed for hierarchically-structured response variables, e.g. the nested logit model. But they have been formally defined for only two or three levels in the hierarchy. Here, we introduce the class of partitioned conditional generalized linear models (PCGLMs) defined for any numbers of levels. The hierarchical structure of these models is fully specified by a partition tree of categories. Using the genericity of the (r,F,Z) specification, the PCGLM can handle nominal, ordinal but also partially-ordered response variables.
We present an approach to deep estimation of discrete conditional probability distributions. Such models have several applications, including generative modeling of audio, image, and video data. Our approach combines two main techniques: dyadic partitioning and graph-based smoothing of the discrete space. By recursively decomposing each dimension into a series of binary splits and smoothing over the resulting distribution using graph-based trend filtering, we impose a strict structure to the model and achieve much higher sample efficiency. We demonstrate the advantages of our model through a s…
We introduce a nonparametric prior on the conditional distribution of a (univariate or multivariate) response given a set of predictors. The prior is constructed in the form of a two-stage generative procedure, which in the first stage recursively partitions the predictor space, and then in the second stage generates the conditional distribution by a multi-scale nonparametric density model on each predictor partition block generated in the first stage. This design allows adaptive smoothing on both the predictor space and the response space, and it results in the full posterior conjugacy of the…
Eliminating the effect of confounding in observational studies typically involves fitting a model for an outcome adjusted for covariates. When, as often, these covariates are high-dimensional, this necessitates the use of sparse estimators such as the Lasso, or other regularisation approaches. Naiive use of such estimators yields confidence intervals for the conditional treatment effect parameter that are not uniformly valid. Moreover, as the number of covariates grows with sample size, correctly specifying a model for the outcome is non-trivial. In this work, we deal with both of these concer…
This paper develops a conformal method to compute prediction intervals for non-parametric regression that can automatically adapt to skewed data. Leveraging black-box machine learning algorithms to estimate the conditional distribution of the outcome using histograms, it translates their output into the shortest prediction intervals with approximate conditional coverage. The resulting prediction intervals provably have marginal coverage in finite samples, while asymptotically achieving conditional coverage and optimal length if the black-box model is consistent. Numerical experiments with simu…
The behavior of the conditional logistic estimator is analyzed under a causal model for two-arm experimental studies with possible non-compliance in which the effect of the treatment is measured by a binary response variable. We show that, when non-compliance may only be observed in the treatment arm, the effect (measured on the logit scale) of the treatment on compliers and that of the control on non-compliers can be identified and consistently estimated under mild conditions. The same does not happen for the effect of the control on compliers. A simple correction of the conditional logistic …
Nonparametric estimation of the conditional expectation $E(Y | U)$ of an outcome $Y$ given a covariate vector $U$ is of primary importance in many statistical applications such as prediction and personalized medicine. In some problems, there is an additional auxiliary variable $Z$ in the training dataset used to construct estimators, but $Z$ is not available for future prediction or selecting patient treatment in personalized medicine. For example, in the training dataset longitudinal outcomes are observed, but only the last outcome $Y$ is concerned in the future prediction or analysis. The lo…
Columbia University（哥伦比亚大学）；UT Dallas（德克萨斯大学达拉斯分校）；UMass Amherst（马萨诸塞大学阿默斯特分校）
Many machine learning tasks can be formulated in terms of predicting structured outputs. In frameworks such as the structured support vector machine (SVM-Struct) and the structured perceptron, discriminative functions are learned by iteratively applying efficient maximum a posteriori (MAP) decoding. However, maximum likelihood estimation (MLE) of probabilistic models over these same structured spaces requires computing partition functions, which is generally intractable. This paper presents a method for learning discrete exponential family models using the Bethe approximation to the MLE. Remar…
Conformal methods create prediction bands that control average coverage under no assumptions besides i.i.d. data. Besides average coverage, one might also desire to control conditional coverage, that is, coverage for every new testing point. However, without strong assumptions, conditional coverage is unachievable. Given this limitation, the literature has focused on methods with asymptotical conditional coverage. In order to obtain this property, these methods require strong conditions on the dependence between the target variable and the features. We introduce two conformal methods based on …
We present a new machine learning approach to estimate personalized treatment effects in the classical potential outcomes framework with binary outcomes. To overcome the problem that both treatment and control outcomes for the same unit are required for supervised learning, we propose surrogate loss functions that incorporate both treatment and control data. The new surrogates yield tighter bounds than the sum of losses for treatment and control groups. A specific choice of loss function, namely a type of hinge loss, yields a minimax support vector machine formulation. The resulting optimizati…
Huawei R&D UK（华为英国研发中心）；University College London（伦敦大学学院）
We present a method for conditional sampling for pre-trained normalizing flows when only part of an observation is available. We derive a lower bound to the conditioning variable log-probability using Schur complement properties in the spirit of Gaussian conditional sampling. Our derivation relies on partitioning flow's domain in such a way that the flow restrictions to subdomains remain bijective, which is crucial for the Schur complement application. Simulation from the variational conditional flow then amends to solving an equality constraint. Our contribution is three-fold: a) we provide d…
Preference-based Conditional Treatment Effects and Policy Learning
Technion – Israel Institute of Technology（技术学院——以色列理工学院）；Inria（法国国家信息与自动化技术研究所）；Inserm（法国国家医学研究院）；Université de Montpellier（蒙彼利埃大学）；Department of Statistics and Operations Research, Tel Aviv University（特拉维夫大学统计与运筹学系）
Causal Discovery with Fewer Conditional Independence Tests
Eric and Wendy Schmidt Center, Broad Institute（埃里克和温迪·施密特中心，布罗德研究所）；Laboratory for Information & Decision Systems, Massachusetts Institute of Technology（麻省理工学院信息与决策系统实验室）
KOF Swiss Economic Institute, ETH Zurich（瑞士经济研究所，苏黎世联邦理工学院）；University of Zurich（苏黎世大学）；LMU Munich（慕尼黑大学）
Learning the cumulative distribution function (CDF) of an outcome variable conditional on a set of features remains challenging, especially in high-dimensional settings. Conditional transformation models provide a semi-parametric approach that allows to model a large class of conditional CDFs without an explicit parametric distribution assumption and with only a few parameters. Existing estimation approaches within this class are, however, either limited in their complexity and applicability to unstructured data sources such as images or text, lack interpretability, or are restricted to certai…
Conditional Distributional Treatment Effects: Doubly Robust Estimation and Testing
Department of Statistics, University of Washington, Seattle, USA（华盛顿大学统计系，美国西雅图）；Departments of Health Care Policy and Statistics, Harvard University, USA（哈佛大学健康政策与统计系，美国）
The goal of feature selection is to identify important features that are relevant to explain an outcome variable. Most of the work in this domain has focused on identifying globally relevant features, which are features that are related to the outcome using evidence across the entire dataset. We study a more fine-grained statistical problem: conditional feature selection, where a feature may be relevant depending on the values of the other features. For example in genetic association studies, variant $A$ could be associated with the phenotype in the entire dataset, but conditioned on variant $…
Outlier detection aims to identify unusual data instances that deviate from expected patterns. The outlier detection is particularly challenging when outliers are context dependent and when they are defined by unusual combinations of multiple outcome variable values. In this paper, we develop and study a new conditional outlier detection approach for multivariate outcome spaces that works by (1) transforming the conditional detection to the outlier detection problem in a new (unconditional) space and (2) defining outlier scores by analyzing the data in the new space. Our approach relies on the…
Icahn School of Medicine at Mount Sinai（西奈山伊坎医学院）；Hasso Plattner Institute for Digital Health at Mount Sinai（西奈山哈索·普拉特纳数字健康研究所）；University of Zurich（苏黎世大学）；Zurich University of Applied Sciences（苏黎世应用科学大学）；Chalmers University of Technology（查尔姆斯理工大学）；Memorial Sloan Kettering Cancer Center（纪念斯隆·凯特琳癌症中心）
An every increasing number of clinical trials features a time-to-event outcome and records non-tabular patient data, such as magnetic resonance imaging or text data in the form of electronic health records. Recently, several neural-network based solutions have been proposed, some of which are binary classifiers. Parametric, distribution-free approaches which make full use of survival time and censoring status have not received much attention. We present deep conditional transformation models (DCTMs) for survival outcomes as a unifying approach to parametric and semiparametric survival analysis…
