Rafael Izbicki

Luca Masserano, Tommaso Dorigo, Rafael Izbicki et al.

Prediction algorithms, such as deep neural networks (DNNs), are used in many domain sciences to directly estimate internal parameters of interest in simulator-based models, especially in settings where the observations include images or complex high-dimensional data. In parallel, modern neural density estimators, such as normalizing flows, are becoming increasingly popular for uncertainty quantification, especially when both parameters and observations are high-dimensional. However, parameter inference is an inverse problem and not a prediction task; thus, an open challenge is to construct conditionally valid and precise confidence regions, with a guaranteed probability of covering the true parameters of the data-generating process, no matter what the (unknown) parameter values are, and without relying on large-sample theory. Many simulator-based inference (SBI) methods are indeed known to produce biased or overly confident parameter regions, yielding misleading uncertainty estimates. This paper presents WALDO, a novel method to construct confidence regions with finite-sample conditional validity by leveraging prediction algorithms or posterior estimators that are currently widely adopted in SBI. WALDO reframes the well-known Wald test statistic, and uses a computationally efficient regression-based machinery for classical Neyman inversion of hypothesis tests. We apply our method to a recent high-energy physics problem, where prediction with DNNs has previously led to estimates with prediction bias. We also illustrate how our approach can correct overly confident posterior regions computed with normalizing flows.

Biprateep Dey, David Zhao, Brett H. Andrews et al.

Key science questions, such as galaxy distance estimation and weather forecasting, often require knowing the full predictive distribution of a target variable $y$ given complex inputs $\mathbf{x}$. Despite recent advances in machine learning and physics-based models, it remains challenging to assess whether an initial model is calibrated for all $\mathbf{x}$, and when needed, to reshape the densities of $y$ toward "instance-wise" calibration. This paper introduces the LADaR (Local Amortized Diagnostics and Reshaping of Conditional Densities) framework and proposes a new computationally efficient algorithm ($\texttt{Cal-PIT}$) that produces interpretable local diagnostics and provides a mechanism for adjusting conditional density estimates (CDEs). $\texttt{Cal-PIT}$ learns a single interpretable local probability--probability map from calibration data that identifies where and how the initial model is miscalibrated across feature space, which can be used to morph CDEs such that they are well-calibrated. We illustrate the LADaR framework on synthetic examples, including probabilistic forecasting from image sequences, akin to predicting storm wind speed from satellite imagery. Our main science application involves estimating the probability density functions of galaxy distances given photometric data, where $\texttt{Cal-PIT}$ achieves better instance-wise calibration than all 11 other literature methods in a benchmark data challenge, demonstrating its utility for next-generation cosmological analyses.

Gilson Y. Shimizu, Rafael Izbicki, Andre C. P. L. F. de Carvalho

A fundamental question on the use of ML models concerns the explanation of their predictions for increasing transparency in decision-making. Although several interpretability methods have emerged, some gaps regarding the reliability of their explanations have been identified. For instance, most methods are unstable (meaning that they give very different explanations with small changes in the data), and do not cope well with irrelevant features (that is, features not related to the label). This article introduces two new interpretability methods, namely VarImp and SupClus, that overcome these issues by using local regressions fits with a weighted distance that takes into account variable importance. Whereas VarImp generates explanations for each instance and can be applied to datasets with more complex relationships, SupClus interprets clusters of instances with similar explanations and can be applied to simpler datasets where clusters can be found. We compare our methods with state-of-the art approaches and show that it yields better explanations according to several metrics, particularly in high-dimensional problems with irrelevant features, as well as when the relationship between features and target is non-linear.

Felipe Maia Polo, Rafael Izbicki, Evanildo Gomes Lacerda et al.

Supervised learning techniques typically assume training data originates from the target population. Yet, in reality, dataset shift frequently arises, which, if not adequately taken into account, may decrease the performance of their predictors. In this work, we propose a novel and flexible framework called DetectShift that quantifies and tests for multiple dataset shifts, encompassing shifts in the distributions of $(X, Y)$, $X$, $Y$, $X|Y$, and $Y|X$. DetectShift equips practitioners with insights into data shifts, facilitating the adaptation or retraining of predictors using both source and target data. This proves extremely valuable when labeled samples in the target domain are limited. The framework utilizes test statistics with the same nature to quantify the magnitude of the various shifts, making results more interpretable. It is versatile, suitable for regression and classification tasks, and accommodates diverse data forms - tabular, text, or image. Experimental results demonstrate the effectiveness of DetectShift in detecting dataset shifts even in higher dimensions.

Alek Fröhlich, Thiago Ramos, Gustavo Cabello et al.

Correctly assessing the malignancy of breast lesions identified during ultrasound examinations is crucial for effective clinical decision-making. However, the current "golden standard" relies on manual BI-RADS scoring by clinicians, often leading to unnecessary biopsies and a significant mental health burden on patients and their families. In this paper, we introduce PersonalizedUS, an interpretable machine learning system that leverages recent advances in conformal prediction to provide precise and personalized risk estimates with local coverage guarantees and sensitivity, specificity, and predictive values above 0.9 across various threshold levels. In particular, we identify meaningful lesion subgroups where distribution-free, model-agnostic conditional coverage holds, with approximately 90% of our prediction sets containing only the ground truth in most lesion subgroups, thus explicitly characterizing for which patients the model is most suitably applied. Moreover, we make available a curated tabular dataset of 1936 biopsied breast lesions from a recent observational multicenter study and benchmark the performance of several state-of-the-art learning algorithms. We also report a successful case study of the deployed system in the same multicenter context. Concrete clinical benefits include up to a 65% reduction in requested biopsies among BI-RADS 4a and 4b lesions, with minimal to no missed cancer cases.

Gabriel O. Assunção, Rafael Izbicki, Marcos O. Prates

Imbalanced datasets present a significant challenge for machine learning models, often leading to biased predictions. To address this issue, data augmentation techniques are widely used in natural language processing (NLP) to generate new samples for the minority class. However, in this paper, we challenge the common assumption that data augmentation is always necessary to improve predictions on imbalanced datasets. Instead, we argue that adjusting the classifier cutoffs without data augmentation can produce similar results to oversampling techniques. Our study provides theoretical and empirical evidence to support this claim. Our findings contribute to a better understanding of the strengths and limitations of different approaches to dealing with imbalanced data, and help researchers and practitioners make informed decisions about which methods to use for a given task.

Gustavo Grivol, Rafael Izbicki, Alex A. Okuno et al.

This paper introduces FlexCodeTS, a new conditional density estimator for time series. FlexCodeTS is a flexible nonparametric conditional density estimator, which can be based on an arbitrary regression method. It is shown that FlexCodeTS inherits the rate of convergence of the chosen regression method. Hence, FlexCodeTS can adapt its convergence by employing the regression method that best fits the structure of data. From an empirical perspective, FlexCodeTS is compared to NNKCDE and GARCH in both simulated and real data. FlexCodeTS is shown to generally obtain the best performance among the selected methods according to either the CDE loss or the pinball loss.

Mateus P. Otto, Rafael Izbicki

Kernel methods provide a flexible and theoretically grounded approach to nonlinear and nonparametric learning. While memory and run-time requirements hinder their applicability to large datasets, many low-rank kernel approximations, such as random Fourier features, were recently developed to scale up such kernel methods. However, these scalable approaches are based on approximations of isotropic kernels, which cannot remove the influence of irrelevant features. In this work, we design random Fourier features for a family of automatic relevance determination (ARD) kernels, and introduce RFFNet, a new large-scale kernel method that learns the kernel relevances' on the fly via first-order stochastic optimization. We present an effective initialization scheme for the method's non-convex objective function, evaluate if hard-thresholding RFFNet's learned relevances yield a sensible rule for variable selection, and perform an extensive ablation study of RFFNet's components. Numerical validation on simulated and real-world data shows that our approach has a small memory footprint and run-time, achieves low prediction error, and effectively identifies relevant features, thus leading to more interpretable solutions. We supply users with an efficient, PyTorch-based library, that adheres to the scikit-learn standard API and code for fully reproducing our results.

Matheus Barreto, Mário de Castro, Thiago R. Ramos et al.

Modern machine learning models can be accurate on average yet still make mistakes that dominate deployment cost. We introduce Locus, a distribution-free wrapper that produces a per-input loss-scale reliability score for a fixed prediction function. Rather than quantifying uncertainty about the label, Locus models the realized loss of the prediction function using any engine that outputs a predictive distribution for the loss given an input. A simple split-calibration step turns this function into a distribution-free interpretable score that is comparable across inputs and can be read as an upper loss level. The score is useful on its own for ranking, and it can optionally be thresholded to obtain a transparent flagging rule with distribution-free control of large-loss events. Experiments across 13 regression benchmarks show that Locus yields effective risk ranking and reduces large-loss frequency compared to standard heuristics.

Tiago Mendonça dos Santos, Rafael Izbicki, Luís Gustavo Esteves

Random Forest ensembles are a strong baseline for tabular prediction tasks, but their reliance on hundreds of deep trees often results in high inference latency and memory demands, limiting deployment in latency-sensitive or resource-constrained environments. We introduce DiNo (Distance with Nodes) and RanBu (Random Bushes), two shallow-forest methods that convert a small set of depth-limited trees into efficient, distance-weighted predictors. DiNo measures cophenetic distances via the most recent common ancestor of observation pairs, while RanBu applies kernel smoothing to Breiman's classical proximity measure. Both approaches operate entirely after forest training: no additional trees are grown, and tuning of the single bandwidth parameter $h$ requires only lightweight matrix-vector operations. Across three synthetic benchmarks and 25 public datasets, RanBu matches or exceeds the accuracy of full-depth random forests-particularly in high-noise settings-while reducing training plus inference time by up to 95\%. DiNo achieves the best bias-variance trade-off in low-noise regimes at a modest computational cost. Both methods extend directly to quantile regression, maintaining accuracy with substantial speed gains. The implementation is available as an open-source R/C++ package at https://github.com/tiagomendonca/dirf. We focus on structured tabular random samples (i.i.d.), leaving extensions to other modalities for future work.

Niccolò Dalmasso, Luca Masserano, David Zhao et al.

Many areas of science rely on simulators that implicitly encode intractable likelihood functions of complex systems. Classical statistical methods are poorly suited for these so-called likelihood-free inference (LFI) settings, especially outside asymptotic and low-dimensional regimes. At the same time, popular LFI methods - such as Approximate Bayesian Computation or more recent machine learning techniques - do not necessarily lead to valid scientific inference because they do not guarantee confidence sets with nominal coverage in general settings. In addition, LFI currently lacks practical diagnostic tools to check the actual coverage of computed confidence sets across the entire parameter space. In this work, we propose a modular inference framework that bridges classical statistics and modern machine learning to provide (i) a practical approach for constructing confidence sets with near finite-sample validity at any value of the unknown parameters, and (ii) interpretable diagnostics for estimating empirical coverage across the entire parameter space. We refer to this framework as likelihood-free frequentist inference (LF2I). Any method that defines a test statistic can leverage LF2I to create valid confidence sets and diagnostics without costly Monte Carlo or bootstrap samples at fixed parameter settings. We study two likelihood-based test statistics (ACORE and BFF) and demonstrate their performance on high-dimensional complex data. Code is available at https://github.com/lee-group-cmu/lf2i.

Tiago Botari, Frederik Hvilshøj, Rafael Izbicki et al.

Most state-of-the-art machine learning algorithms induce black-box models, preventing their application in many sensitive domains. Hence, many methodologies for explaining machine learning models have been proposed to address this problem. In this work, we introduce strategies to improve local explanations taking into account the distribution of the data used to train the black-box models. We show that our approach, MeLIME, produces more meaningful explanations compared to other techniques over different ML models, operating on various types of data. MeLIME generalizes the LIME method, allowing more flexible perturbation sampling and the use of different local interpretable models. Additionally, we introduce modifications to standard training algorithms of local interpretable models fostering more robust explanations, even allowing the production of counterfactual examples. To show the strengths of the proposed approach, we include experiments on tabular data, images, and text; all showing improved explanations. In particular, MeLIME generated more meaningful explanations on the MNIST dataset than methods such as GuidedBackprop, SmoothGrad, and Layer-wise Relevance Propagation. MeLIME is available on https://github.com/tiagobotari/melime.

Niccolò Dalmasso, Rafael Izbicki, Ann B. Lee

Parameter estimation, statistical tests and confidence sets are the cornerstones of classical statistics that allow scientists to make inferences about the underlying process that generated the observed data. A key question is whether one can still construct hypothesis tests and confidence sets with proper coverage and high power in a so-called likelihood-free inference (LFI) setting; that is, a setting where the likelihood is not explicitly known but one can forward-simulate observable data according to a stochastic model. In this paper, we present $\texttt{ACORE}$ (Approximate Computation via Odds Ratio Estimation), a frequentist approach to LFI that first formulates the classical likelihood ratio test (LRT) as a parametrized classification problem, and then uses the equivalence of tests and confidence sets to build confidence regions for parameters of interest. We also present a goodness-of-fit procedure for checking whether the constructed tests and confidence regions are valid. $\texttt{ACORE}$ is based on the key observation that the LRT statistic, the rejection probability of the test, and the coverage of the confidence set are conditional distribution functions which often vary smoothly as a function of the parameters of interest. Hence, instead of relying solely on samples simulated at fixed parameter settings (as is the convention in standard Monte Carlo solutions), one can leverage machine learning tools and data simulated in the neighborhood of a parameter to improve estimates of quantities of interest. We demonstrate the efficacy of $\texttt{ACORE}$ with both theoretical and empirical results. Our implementation is available on Github.

Niccolò Dalmasso, Taylor Pospisil, Ann B. Lee et al.

It is well known in astronomy that propagating non-Gaussian prediction uncertainty in photometric redshift estimates is key to reducing bias in downstream cosmological analyses. Similarly, likelihood-free inference approaches, which are beginning to emerge as a tool for cosmological analysis, require a characterization of the full uncertainty landscape of the parameters of interest given observed data. However, most machine learning (ML) or training-based methods with open-source software target point prediction or classification, and hence fall short in quantifying uncertainty in complex regression and parameter inference settings. As an alternative to methods that focus on predicting the response (or parameters) $\mathbf{y}$ from features $\mathbf{x}$, we provide nonparametric conditional density estimation (CDE) tools for approximating and validating the entire probability density function (PDF) $\mathrm{p}(\mathbf{y}|\mathbf{x})$ of $\mathbf{y}$ given (i.e., conditional on) $\mathbf{x}$. As there is no one-size-fits-all CDE method, the goal of this work is to provide a comprehensive range of statistical tools and open-source software for nonparametric CDE and method assessment which can accommodate different types of settings and be easily fit to the problem at hand. Specifically, we introduce four CDE software packages in $\texttt{Python}$ and $\texttt{R}$ based on ML prediction methods adapted and optimized for CDE: $\texttt{NNKCDE}$, $\texttt{RFCDE}$, $\texttt{FlexCode}$, and $\texttt{DeepCDE}$. Furthermore, we present the $\texttt{cdetools}$ package, which includes functions for computing a CDE loss function for tuning and assessing the quality of individual PDFs, along with diagnostic functions. We provide sample code in $\texttt{Python}$ and $\texttt{R}$ as well as examples of applications to photometric redshift estimation and likelihood-free cosmological inference via CDE.

Rafael Izbicki, Pedro L. C. Rodrigues

Conditional density estimation (CDE) - recovering the full conditional distribution of a response given tabular covariates - is essential in settings with heteroscedasticity, multimodality, or asymmetric uncertainty. Recent tabular foundation models, such as TabPFN and TabICL, naturally produce predictive distributions, but their effectiveness as general-purpose CDE methods has not been systematically evaluated, unlike their performance for point prediction, which is well studied. We benchmark three tabular foundation model variants against a diverse set of parametric, tree-based, and neural CDE baselines on 39 real-world datasets, across training sizes from 50 to 20,000, using six metrics covering density accuracy, calibration, and computation time. Across all sample sizes, foundation models achieve the best CDE loss, log-likelihood, and CRPS on the large majority of datasets tested. Calibration is competitive at small sample sizes but, for some metrics and datasets, lags behind task-specific neural baselines at larger sample sizes, suggesting that post-hoc recalibration may be a valuable complement. In a photometric redshift case study using SDSS DR18, TabPFN exposed to 50,000 training galaxies outperforms all baselines trained on the full 500,000-galaxy dataset. Taken together, these results establish tabular foundation models as strong off-the-shelf conditional density estimators.

Vagner Santos, Victor Coscrato, Luben Cabezas et al.

Gradient-boosted decision trees are among the strongest off-the-shelf predictors for tabular regression, but point predictions alone do not quantify uncertainty. Conformal prediction provides distribution-free marginal coverage, yet split conformal uses a single global residual quantile and can be poorly adaptive under heteroscedasticity. Methods that improve adaptivity typically fit auxiliary nuisance models or introduce additional data splits/partitions to learn the conformal score, increasing cost and reducing data efficiency. We propose LoBoost, a model-native local conformal method that reuses the fitted ensemble's leaf structure to define multiscale calibration groups. Each input is encoded by its sequence of visited leaves; at resolution level k, we group points by matching prefixes of leaf indices across the first k trees and calibrate residual quantiles within each group. LoBoost requires no retraining, auxiliary models, or extra splitting beyond the standard train/calibration split. Experiments show competitive interval quality, improved test MSE on most datasets, and large calibration speedups.

Elizabeth Cucuzzella, Rafael Izbicki, Ann B. Lee

Forecast systems in science and technology are increasingly moving beyond point prediction toward methods that produce full predictive distributions of future outcomes y, conditional on high-dimensional and complex sequences of inputs x. However, even when forecast systems provide a full predictive distribution, the result is rarely calibrated with respect to all x and y. The estimated density can be especially unreliable in low-frequency or out-of-distribution regimes, where accurate uncertainty quantification and a means for human experts to verify results are most needed to establish trust in models. In this paper, we take an initial predictive distribution as given and treat it as a useful but potentially misspecified base model. WE then introduce diagnostic transport maps, covariate-dependent probability-to-probability maps that quantify how the base model's probabilities should be adjusted to better match the true conditional distribution of calibration data. At deployment, these maps provide the user with real-time local diagnostics that reveal where the model fails and how it fails (including bias, dispersion, skewness, and tail errors), while also producing a recalibrated predictive distribution through a simple composition with the base model. We apply diagnostic transport maps to short-term tropical cyclone intensity forecasting and show that an easy-to-fit parametric version identifies evolutionary modes associated with local miscalibration and improves the predictive performance for rare events, including 24-hour rapid intensity change, as compared to the operational forecasts of the National Hurricane Center.

Luben M. C. Cabezas, Mateus P. Otto, Rafael Izbicki et al.

Predictive models make mistakes. Hence, there is a need to quantify the uncertainty associated with their predictions. Conformal inference has emerged as a powerful tool to create statistically valid prediction regions around point predictions, but its naive application to regression problems yields non-adaptive regions. New conformal scores, often relying upon quantile regressors or conditional density estimators, aim to address this limitation. Although they are useful for creating prediction bands, these scores are detached from the original goal of quantifying the uncertainty around an arbitrary predictive model. This paper presents a new, model-agnostic family of methods to calibrate prediction intervals for regression problems with local coverage guarantees. Our approach is based on pursuing the coarsest partition of the feature space that approximates conditional coverage. We create this partition by training regression trees and Random Forests on conformity scores. Our proposal is versatile, as it applies to various conformity scores and prediction settings and demonstrates superior scalability and performance compared to established baselines in simulated and real-world datasets. We provide a Python package clover that implements our methods using the standard scikit-learn interface.

Victor Dheur, Tanguy Bosser, Rafael Izbicki et al.

Sequences of labeled events observed at irregular intervals in continuous time are ubiquitous across various fields. Temporal Point Processes (TPPs) provide a mathematical framework for modeling these sequences, enabling inferences such as predicting the arrival time of future events and their associated label, called mark. However, due to model misspecification or lack of training data, these probabilistic models may provide a poor approximation of the true, unknown underlying process, with prediction regions extracted from them being unreliable estimates of the underlying uncertainty. This paper develops more reliable methods for uncertainty quantification in neural TPP models via the framework of conformal prediction. A primary objective is to generate a distribution-free joint prediction region for an event's arrival time and mark, with a finite-sample marginal coverage guarantee. A key challenge is to handle both a strictly positive, continuous response and a categorical response, without distributional assumptions. We first consider a simple but conservative approach that combines individual prediction regions for the event's arrival time and mark. Then, we introduce a more effective method based on bivariate highest density regions derived from the joint predictive density of arrival times and marks. By leveraging the dependencies between these two variables, this method excludes unlikely combinations of the two, resulting in sharper prediction regions while still attaining the pre-specified coverage level. We also explore the generation of individual univariate prediction regions for events' arrival times and marks through conformal regression and classification techniques. Moreover, we evaluate the stronger notion of conditional coverage. Finally, through extensive experimentation on both simulated and real-world datasets, we assess the validity and efficiency of these methods.

Luben M. C. Cabezas, Vagner S. Santos, Thiago R. Ramos et al.

Conformal prediction methods create prediction bands with distribution-free guarantees but do not explicitly capture epistemic uncertainty, which can lead to overconfident predictions in data-sparse regions. Although recent conformal scores have been developed to address this limitation, they are typically designed for specific tasks, such as regression or quantile regression. Moreover, they rely on particular modeling choices for epistemic uncertainty, restricting their applicability. We introduce $\texttt{EPICSCORE}$, a model-agnostic approach that enhances any conformal score by explicitly integrating epistemic uncertainty. Leveraging Bayesian techniques such as Gaussian Processes, Monte Carlo Dropout, or Bayesian Additive Regression Trees, $\texttt{EPICSCORE}$ adaptively expands predictive intervals in regions with limited data while maintaining compact intervals where data is abundant. As with any conformal method, it preserves finite-sample marginal coverage. Additionally, it also achieves asymptotic conditional coverage. Experiments demonstrate its good performance compared to existing methods. Designed for compatibility with any Bayesian model, but equipped with distribution-free guarantees, $\texttt{EPICSCORE}$ provides a general-purpose framework for uncertainty quantification in prediction problems.

James Carzon, Luca Masserano, Joshua D. Ingram et al.

Generative artificial intelligence (AI) excels at producing complex data structures (text, images, videos) by learning patterns from training examples. Across scientific disciplines, researchers are now applying generative models to ``inverse problems'' to infer hidden parameters from observed data. While these methods can handle intractable models and large-scale studies, they can also produce biased or overconfident conclusions. We present a solution with Frequentist-Bayes (FreB), a mathematically rigorous protocol that reshapes AI-generated probability distributions into confidence regions that consistently include true parameters with the expected probability, while achieving minimum size when training and target data align. We demonstrate FreB's effectiveness by tackling diverse case studies in the physical sciences: identifying unknown sources under dataset shift, reconciling competing theoretical models, and mitigating selection bias and systematics in observational studies. By providing validity guarantees with interpretable diagnostics, FreB enables trustworthy scientific inference across fields where direct likelihood evaluation remains impossible or prohibitively expensive.

Luben M. C. Cabezas, Vagner S. Santos, Thiago R. Ramos et al.

Current experimental scientists have been increasingly relying on simulation-based inference (SBI) to invert complex non-linear models with intractable likelihoods. However, posterior approximations obtained with SBI are often miscalibrated, causing credible regions to undercover true parameters. We develop $\texttt{CP4SBI}$, a model-agnostic conformal calibration framework that constructs credible sets with local Bayesian coverage. Our two proposed variants, namely local calibration via regression trees and CDF-based calibration, enable finite-sample local coverage guarantees for any scoring function, including HPD, symmetric, and quantile-based regions. Experiments on widely used SBI benchmarks demonstrate that our approach improves the quality of uncertainty quantification for neural posterior estimators using both normalizing flows and score-diffusion modeling.

Luca Masserano, Alex Shen, Michele Doro et al.

An open scientific challenge is how to classify events with reliable measures of uncertainty, when we have a mechanistic model of the data-generating process but the distribution over both labels and latent nuisance parameters is different between train and target data. We refer to this type of distributional shift as generalized label shift (GLS). Direct classification using observed data $\mathbf{X}$ as covariates leads to biased predictions and invalid uncertainty estimates of labels $Y$. We overcome these biases by proposing a new method for robust uncertainty quantification that casts classification as a hypothesis testing problem under nuisance parameters. The key idea is to estimate the classifier's receiver operating characteristic (ROC) across the entire nuisance parameter space, which allows us to devise cutoffs that are invariant under GLS. Our method effectively endows a pre-trained classifier with domain adaptation capabilities and returns valid prediction sets while maintaining high power. We demonstrate its performance on two challenging scientific problems in biology and astroparticle physics with data from realistic mechanistic models.

Milene Regina dos Santos, Rafael Izbicki

Regression methods assume that accurate labels are available for training. However, in certain scenarios, obtaining accurate labels may not be feasible, and relying on multiple specialists with differing opinions becomes necessary. Existing approaches addressing noisy labels often impose restrictive assumptions on the regression function. In contrast, this paper presents a novel, more flexible approach. Our method consists of two steps: estimating each labeler's expertise and combining their opinions using learned weights. We then regress the weighted average against the input features to build the prediction model. The proposed method is formally justified and empirically demonstrated to outperform existing techniques on simulated and real data. Furthermore, its flexibility enables the utilization of any machine learning technique in both steps. In summary, this method offers a simple, fast, and effective solution for training regression models with noisy labels derived from diverse expert opinions.

Gilson Shimizu, Rafael Izbicki, Denis Valle

The Latent Dirichlet Allocation (LDA) model is a popular method for creating mixed-membership clusters. Despite having been originally developed for text analysis, LDA has been used for a wide range of other applications. We propose a new formulation for the LDA model which incorporates covariates. In this model, a negative binomial regression is embedded within LDA, enabling straight-forward interpretation of the regression coefficients and the analysis of the quantity of cluster-specific elements in each sampling units (instead of the analysis being focused on modeling the proportion of each cluster, as in Structural Topic Models). We use slice sampling within a Gibbs sampling algorithm to estimate model parameters. We rely on simulations to show how our algorithm is able to successfully retrieve the true parameter values and the ability to make predictions for the abundance matrix using the information given by the covariates. The model is illustrated using real data sets from three different areas: text-mining of Coronavirus articles, analysis of grocery shopping baskets, and ecology of tree species on Barro Colorado Island (Panama). This model allows the identification of mixed-membership clusters in discrete data and provides inference on the relationship between covariates and the abundance of these clusters.

Trey McNeely, Galen Vincent, Kimberly M. Wood et al.

Our goal is to quantify whether, and if so how, spatio-temporal patterns in tropical cyclone (TC) satellite imagery signal an upcoming rapid intensity change event. To address this question, we propose a new nonparametric test of association between a time series of images and a series of binary event labels. We ask whether there is a difference in distribution between (dependent but identically distributed) 24-h sequences of images preceding an event versus a non-event. By rewriting the statistical test as a regression problem, we leverage neural networks to infer modes of structural evolution of TC convection that are representative of the lead-up to rapid intensity change events. Dependencies between nearby sequences are handled by a bootstrap procedure that estimates the marginal distribution of the label series. We prove that type I error control is guaranteed as long as the distribution of the label series is well-estimated, which is made easier by the extensive historical data for binary TC event labels. We show empirical evidence that our proposed method identifies archetypes of infrared imagery associated with elevated rapid intensification risk, typically marked by deep or deepening core convection over time. Such results provide a foundation for improved forecasts of rapid intensification.

Luben M. C. Cabezas, Rafael Izbicki, Rafael B. Stern

We propose methods for the analysis of hierarchical clustering that fully use the multi-resolution structure provided by a dendrogram. Specifically, we propose a loss for choosing between clustering methods, a feature importance score and a graphical tool for visualizing the segmentation of features in a dendrogram. Current approaches to these tasks lead to loss of information since they require the user to generate a single partition of the instances by cutting the dendrogram at a specified level. Our proposed methods, instead, use the full structure of the dendrogram. The key insight behind the proposed methods is to view a dendrogram as a phylogeny. This analogy permits the assignment of a feature value to each internal node of a tree through an evolutionary model. Real and simulated datasets provide evidence that our proposed framework has desirable outcomes and gives more insights than state-of-art approaches. We provide an R package that implements our methods.

Biprateep Dey, Jeffrey A. Newman, Brett H. Andrews et al.

Many astrophysical analyses depend on estimates of redshifts (a proxy for distance) determined from photometric (i.e., imaging) data alone. Inaccurate estimates of photometric redshift uncertainties can result in large systematic errors. However, probability distribution outputs from many photometric redshift methods do not follow the frequentist definition of a Probability Density Function (PDF) for redshift -- i.e., the fraction of times the true redshift falls between two limits $z_{1}$ and $z_{2}$ should be equal to the integral of the PDF between these limits. Previous works have used the global distribution of Probability Integral Transform (PIT) values to re-calibrate PDFs, but offsetting inaccuracies in different regions of feature space can conspire to limit the efficacy of the method. We leverage a recently developed regression technique that characterizes the local PIT distribution at any location in feature space to perform a local re-calibration of photometric redshift PDFs. Though we focus on an example from astrophysics, our method can produce PDFs which are calibrated at all locations in feature space for any use case.

Trey McNeely, Galen Vincent, Rafael Izbicki et al.

Tropical cyclone (TC) intensity forecasts are issued by human forecasters who evaluate spatio-temporal observations (e.g., satellite imagery) and model output (e.g., numerical weather prediction, statistical models) to produce forecasts every 6 hours. Within these time constraints, it can be challenging to draw insight from such data. While high-capacity machine learning methods are well suited for prediction problems with complex sequence data, extracting interpretable scientific information with such methods is difficult. Here we leverage powerful AI prediction algorithms and classical statistical inference to identify patterns in the evolution of TC convective structure leading up to the rapid intensification of a storm, hence providing forecasters and scientists with key insight into TC behavior.

Rafael Izbicki, Gilson Shimizu, Rafael B. Stern

Conformal methods create prediction bands that control average coverage assuming solely i.i.d. data. Although the literature has mostly focused on prediction intervals, more general regions can often better represent uncertainty. For instance, a bimodal target is better represented by the union of two intervals. Such prediction regions are obtained by CD-split , which combines the split method and a data-driven partition of the feature space which scales to high dimensions. CD-split however contains many tuning parameters, and their role is not clear. In this paper, we provide new insights on CD-split by exploring its theoretical properties. In particular, we show that CD-split converges asymptotically to the oracle highest predictive density set and satisfies local and asymptotic conditional validity. We also present simulations that show how to tune CD-split. Finally, we introduce HPD-split, a variation of CD-split that requires less tuning, and show that it shares the same theoretical guarantees as CD-split. In a wide variety of our simulations, CD-split and HPD-split have better conditional coverage and yield smaller prediction regions than other methods.

Rafael Izbicki, Gilson T. Shimizu, Rafael B. Stern

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 conditional density estimators that do not depend on this type of assumption to obtain asymptotic conditional coverage: Dist-split and CD-split. While Dist-split asymptotically obtains optimal intervals, which are easier to interpret than general regions, CD-split obtains optimal size regions, which are smaller than intervals. CD-split also obtains local coverage by creating a data-driven partition of the feature space that scales to high-dimensional settings and by generating prediction bands locally on the partition elements. In a wide variety of simulated scenarios, our methods have a better control of conditional coverage and have smaller length than previously proposed methods.

Victor Coscrato, Marco Henrique de Almeida Inácio, Tiago Botari et al.

An important feature of successful supervised machine learning applications is to be able to explain the predictions given by the regression or classification model being used. However, most state-of-the-art models that have good predictive power lead to predictions that are hard to interpret. Thus, several model-agnostic interpreters have been developed recently as a way of explaining black-box classifiers. In practice, using these methods is a slow process because a novel fitting is required for each new testing instance, and several non-trivial choices must be made. We develop NLS (neural local smoother), a method that is complex enough to give good predictions, and yet gives solutions that are easy to be interpreted without the need of using a separate interpreter. The key idea is to use a neural network that imposes a local linear shape to the output layer. We show that NLS leads to predictive power that is comparable to state-of-the-art machine learning models, and yet is easier to interpret.

Marco Henrique de Almeida Inácio, Rafael Izbicki, Bálint Gyires-Tóth

Given two distinct datasets, an important question is if they have arisen from the the same data generating function or alternatively how their data generating functions diverge from one another. In this paper, we introduce an approach for measuring the distance between two datasets with high dimensionality using variational autoencoders. This approach is augmented by a permutation hypothesis test in order to check the hypothesis that the data generating distributions are the same within a significance level. We evaluate both the distance measurement and hypothesis testing approaches on generated and on public datasets. According to the results the proposed approach can be used for data exploration (e.g. by quantifying the discrepancy/separability between categories of images), which can be particularly useful in the early phases of the pipeline of most machine learning projects.

Marco Henrique de Almeida Inácio, Rafael Izbicki, Rafael Bassi Stern

Conditional independence testing is a key problem required by many machine learning and statistics tools. In particular, it is one way of evaluating the usefulness of some features on a supervised prediction problem. We propose a novel conditional independence test in a predictive setting, and show that it achieves better power than competing approaches in several settings. Our approach consists in deriving a p-value using a permutation test where the predictive power using the unpermuted dataset is compared with the predictive power of using dataset where the feature(s) of interest are permuted. We conclude that the method achives sensible results on simulated and real datasets.

Tiago Botari, Rafael Izbicki, Andre C. P. L. F. de Carvalho

As machine learning becomes an important part of many real world applications affecting human lives, new requirements, besides high predictive accuracy, become important. One important requirement is transparency, which has been associated with model interpretability. Many machine learning algorithms induce models difficult to interpret, named black box. Moreover, people have difficulty to trust models that cannot be explained. In particular for machine learning, many groups are investigating new methods able to explain black box models. These methods usually look inside the black models to explain their inner work. By doing so, they allow the interpretation of the decision making process used by black box models. Among the recently proposed model interpretation methods, there is a group, named local estimators, which are designed to explain how the label of particular instance is predicted. For such, they induce interpretable models on the neighborhood of the instance to be explained. Local estimators have been successfully used to explain specific predictions. Although they provide some degree of model interpretability, it is still not clear what is the best way to implement and apply them. Open questions include: how to best define the neighborhood of an instance? How to control the trade-off between the accuracy of the interpretation method and its interpretability? How to make the obtained solution robust to small variations on the instance to be explained? To answer to these questions, we propose and investigate two strategies: (i) using data instance properties to provide improved explanations, and (ii) making sure that the neighborhood of an instance is properly defined by taking the geometry of the domain of the feature space into account. We evaluate these strategies in a regression task and present experimental results that show that they can improve local explanations.

Victor Coscrato, Marco Henrique de Almeida Inácio, Rafael Izbicki

Stacking methods improve the prediction performance of regression models. A simple way to stack base regressions estimators is by combining them linearly, as done by \citet{breiman1996stacked}. Even though this approach is useful from an interpretative perspective, it often does not lead to high predictive power. We propose the NN-Stacking method (NNS), which generalizes Breiman's method by allowing the linear parameters to vary with input features. This improvement enables NNS to take advantage of the fact that distinct base models often perform better at different regions of the feature space. Our method uses neural networks to estimate the stacking coefficients. We show that while our approach keeps the interpretative features of Breiman's method at a local level, it leads to better predictive power, especially in datasets with large sample sizes.

Niccolò Dalmasso, Ann B. Lee, Rafael Izbicki et al.

Complex phenomena in engineering and the sciences are often modeled with computationally intensive feed-forward simulations for which a tractable analytic likelihood does not exist. In these cases, it is sometimes necessary to estimate an approximate likelihood or fit a fast emulator model for efficient statistical inference; such surrogate models include Gaussian synthetic likelihoods and more recently neural density estimators such as autoregressive models and normalizing flows. To date, however, there is no consistent way of quantifying the quality of such a fit. Here we propose a statistical framework that can distinguish any arbitrary misspecified model from the target likelihood, and that in addition can identify with statistical confidence the regions of parameter as well as feature space where the fit is inadequate. Our validation method applies to settings where simulations are extremely costly and generated in batches or "ensembles" at fixed locations in parameter space. At the heart of our approach is a two-sample test that quantifies the quality of the fit at fixed parameter values, and a global test that assesses goodness-of-fit across simulation parameters. While our general framework can incorporate any test statistic or distance metric, we specifically argue for a new two-sample test that can leverage any regression method to attain high power and provide diagnostics in complex data settings.

Afonso Fernandes Vaz, Rafael Izbicki, Rafael Bassi Stern

The quantification problem consists of determining the prevalence of a given label in a target population. However, one often has access to the labels in a sample from the training population but not in the target population. A common assumption in this situation is that of prior probability shift, that is, once the labels are known, the distribution of the features is the same in the training and target populations. In this paper, we derive a new lower bound for the risk of the quantification problem under the prior shift assumption. Complementing this lower bound, we present a new approximately minimax class of estimators, ratio estimators, which generalize several previous proposals in the literature. Using a weaker version of the prior shift assumption, which can be tested, we show that ratio estimators can be used to build confidence intervals for the quantification problem. We also extend the ratio estimator so that it can: (i) incorporate labels from the target population, when they are available and (ii) estimate how the prevalence of positive labels varies according to a function of certain covariates.

Rafael Izbicki, Ann B. Lee, Taylor Pospisil

Approximate Bayesian Computation (ABC) is typically used when the likelihood is either unavailable or intractable but where data can be simulated under different parameter settings using a forward model. Despite the recent interest in ABC, high-dimensional data and costly simulations still remain a bottleneck in some applications. There is also no consensus as to how to best assess the performance of such methods without knowing the true posterior. We show how a nonparametric conditional density estimation (CDE) framework, which we refer to as ABC-CDE, help address three nontrivial challenges in ABC: (i) how to efficiently estimate the posterior distribution with limited simulations and different types of data, (ii) how to tune and compare the performance of ABC and related methods in estimating the posterior itself, rather than just certain properties of the density, and (iii) how to efficiently choose among a large set of summary statistics based on a CDE surrogate loss. We provide theoretical and empirical evidence that justify ABC-CDE procedures that {\em directly} estimate and assess the posterior based on an initial ABC sample, and we describe settings where standard ABC and regression-based approaches are inadequate.

Rafael Izbicki, Ann B. Lee

There is a growing demand for nonparametric conditional density estimators (CDEs) in fields such as astronomy and economics. In astronomy, for example, one can dramatically improve estimates of the parameters that dictate the evolution of the Universe by working with full conditional densities instead of regression (i.e., conditional mean) estimates. More generally, standard regression falls short in any prediction problem where the distribution of the response is more complex with multi-modality, asymmetry or heteroscedastic noise. Nevertheless, much of the work on high-dimensional inference concerns regression and classification only, whereas research on density estimation has lagged behind. Here we propose FlexCode, a fully nonparametric approach to conditional density estimation that reformulates CDE as a non-parametric orthogonal series problem where the expansion coefficients are estimated by regression. By taking such an approach, one can efficiently estimate conditional densities and not just expectations in high dimensions by drawing upon the success in high-dimensional regression. Depending on the choice of regression procedure, our method can adapt to a variety of challenging high-dimensional settings with different structures in the data (e.g., a large number of irrelevant components and nonlinear manifold structure) as well as different data types (e.g., functional data, mixed data types and sample sets). We study the theoretical and empirical performance of our proposed method, and we compare our approach with traditional conditional density estimators on simulated as well as real-world data, such as photometric galaxy data, Twitter data, and line-of-sight velocities in a galaxy cluster.

Ann B. Lee, Rafael Izbicki

A key question in modern statistics is how to make fast and reliable inferences for complex, high-dimensional data. While there has been much interest in sparse techniques, current methods do not generalize well to data with nonlinear structure. In this work, we present an orthogonal series estimator for predictors that are complex aggregate objects, such as natural images, galaxy spectra, trajectories, and movies. Our series approach ties together ideas from kernel machine learning, and Fourier methods. We expand the unknown regression on the data in terms of the eigenfunctions of a kernel-based operator, and we take advantage of orthogonality of the basis with respect to the underlying data distribution, P, to speed up computations and tuning of parameters. If the kernel is appropriately chosen, then the eigenfunctions adapt to the intrinsic geometry and dimension of the data. We provide theoretical guarantees for a radial kernel with varying bandwidth, and we relate smoothness of the regression function with respect to P to sparsity in the eigenbasis. Finally, using simulated and real-world data, we systematically compare the performance of the spectral series approach with classical kernel smoothing, k-nearest neighbors regression, kernel ridge regression, and state-of-the-art manifold and local regression methods.

Rafael Izbicki, Rafael Bassi Stern

Experts classifying data are often imprecise. Recently, several models have been proposed to train classifiers using the noisy labels generated by these experts. How to choose between these models? In such situations, the true labels are unavailable. Thus, one cannot perform model selection using the standard versions of methods such as empirical risk minimization and cross validation. In order to allow model selection, we present a surrogate loss and provide theoretical guarantees that assure its consistency. Next, we discuss how this loss can be used to tune a penalization which introduces sparsity in the parameters of a traditional class of models. Sparsity provides more parsimonious models and can avoid overfitting. Nevertheless, it has seldom been discussed in the context of noisy labels due to the difficulty in model selection and, therefore, in choosing tuning parameters. We apply these techniques to several sets of simulated and real data.