arXiv Statistics @arxiv_stats@qoto.org

The notion of group invariance helps neural networks in recognizing patterns and features under geometric transformations. Indeed, it has been shown that group invariance can largely improve deep learning performances in practice, where such transformations are very common. This research studies affine invariance on continuous-domain convolutional neural networks. Despite other research considering isometric invariance or similarity invariance, we focus on the full structure of affine transforms generated by the generalized linear group $\mathrm{GL}_2(\mathbb{R})$. We introduce a new criterion to assess the similarity of two input signals under affine transformations. Then, unlike conventional methods that involve solving complex optimization problems on the Lie group $G_2$, we analyze the convolution of lifted signals and compute the corresponding integration over $G_2$. In sum, our research could eventually extend the scope of geometrical transformations that practical deep-learning pipelines can handle.

In this paper, we address the problem of dynamic network embedding, that is, representing the nodes of a dynamic network as evolving vectors within a low-dimensional space. While the field of static network embedding is wide and established, the field of dynamic network embedding is comparatively in its infancy. We propose that a wide class of established static network embedding methods can be used to produce interpretable and powerful dynamic network embeddings when they are applied to the dilated unfolded adjacency matrix. We provide a theoretical guarantee that, regardless of embedding dimension, these unfolded methods will produce stable embeddings, meaning that nodes with identical latent behaviour will be exchangeable, regardless of their position in time or space. We additionally define a hypothesis testing framework which can be used to evaluate the quality of a dynamic network embedding by testing for planted structure in simulated networks. Using this, we demonstrate that, even in trivial cases, unstable methods are often either conservative or encode incorrect structure. In contrast, we demonstrate that our suite of stable unfolded methods are not only more interpretable but also more powerful in comparison to their unstable counterparts.

Dietary intake data are routinely drawn upon to explore diet-health relationships. However, these data are often subject to measurement error, distorting the true relationships. Beyond measurement error, there are likely complex synergistic and sometimes antagonistic interactions between different dietary components, complicating the relationships between diet and health outcomes. Flexible models are required to capture the nuance that these complex interactions introduce. This complexity makes research on diet-health relationships an appealing candidate for the application of machine learning techniques, and in particular, neural networks. Neural networks are computational models that are able to capture highly complex, nonlinear relationships so long as sufficient data are available. While these models have been applied in many domains, the impacts of measurement error on the performance of predictive modeling has not been systematically investigated. However, dietary intake data are typically collected using self-report methods and are prone to large amounts of measurement error. In this work, we demonstrate the ways in which measurement error erodes the performance of neural networks, and illustrate the care that is required for leveraging these models in the presence of error. We demonstrate the role that sample size and replicate measurements play on model performance, indicate a motivation for the investigation of transformations to additivity, and illustrate the caution required to prevent model overfitting. While the past performance of neural networks across various domains make them an attractive candidate for examining diet-health relationships, our work demonstrates that substantial care and further methodological development are both required to observe increased predictive performance when applying these techniques, compared to more traditional statistical procedures.

Electronic health records contain valuable information for monitoring patients' health trajectories over time. Disease progression models have been developed to understand the underlying patterns and dynamics of diseases using these data as sequences. However, analyzing temporal data from EHRs is challenging due to the variability and irregularities present in medical records. We propose a Markovian generative model of treatments developed to (i) model the irregular time intervals between medical events; (ii) classify treatments into subtypes based on the patient sequence of medical events and the time intervals between them; and (iii) segment treatments into subsequences of disease progression patterns. We assume that sequences have an associated structure of latent variables: a latent class representing the different subtypes of treatments; and a set of latent stages indicating the phase of progression of the treatments. We use the Expectation-Maximization algorithm to learn the model, which is efficiently solved with a dynamic programming-based method. Various parametric models have been employed to model the time intervals between medical events during the learning process, including the geometric, exponential, and Weibull distributions. The results demonstrate the effectiveness of our model in recovering the underlying model from data and accurately modeling the irregular time intervals between medical actions.

We construct a counterexample to the injectivity conjecture of Masarotto et al (2018). Namely, we construct a class of examples of injective covariance operators on an infinite-dimensional separable Hilbert space for which the Bures--Wasserstein barycentre is highly non injective -- it has a kernel of infinite dimension.

Research intended to estimate the effect of an action, like in randomized trials, often do not have random samples of the intended target population. Instead, estimates can be transported to the desired target population. Methods for transporting between populations are often premised on a positivity assumption, such that all relevant covariate patterns in one population are also present in the other. However, eligibility criteria, particularly in the case of trials, can result in violations of positivity. To address nonpositivity, a synthesis of statistical and mechanistic models was previously proposed in the context of violations by a single binary covariate. Here, we extend the synthesis approach for positivity violations with a continuous covariate. For estimation, two novel augmented inverse probability weighting estimators are proposed, with one based on estimating the parameters of a marginal structural model and the other based on estimating the conditional average causal effect. Both estimators are compared to other common approaches to address nonpositivity via a simulation study. Finally, the competing approaches are illustrated with an example in the context of two-drug versus one-drug antiretroviral therapy on CD4 T cell counts among women with HIV.

We propose a bootstrap-based test to detect a mean shift in a sequence of high-dimensional observations with unknown time-varying heteroscedasticity. The proposed test builds on the U-statistic based approach in Wang et al. (2022), targets a dense alternative, and adopts a wild bootstrap procedure to generate critical values. The bootstrap-based test is free of tuning parameters and is capable of accommodating unconditional time varying heteroscedasticity in the high-dimensional observations, as demonstrated in our theory and simulations. Theoretically, we justify the bootstrap consistency by using the recently proposed unconditional approach in Bucher and Kojadinovic (2019). Extensions to testing for multiple change-points and estimation using wild binary segmentation are also presented. Numerical simulations demonstrate the robustness of the proposed testing and estimation procedures with respect to different kinds of time-varying heteroscedasticity.

Orthogonal meta-learners, such as DR-learner, R-learner and IF-learner, are increasingly used to estimate conditional average treatment effects. They improve convergence rates relative to na\"ıve meta-learners (e.g., T-, S- and X-learner) through de-biasing procedures that involve applying standard learners to specifically transformed outcome data. This leads them to disregard the possibly constrained outcome space, which can be particularly problematic for dichotomous outcomes: these typically get transformed to values that are no longer constrained to the unit interval, making it difficult for standard learners to guarantee predictions within the unit interval. To address this, we construct orthogonal meta-learners for the prediction of counterfactual outcomes which respect the outcome space. As such, the obtained i-learner or imputation-learner is more generally expected to outperform existing learners, even when the outcome is unconstrained, as we confirm empirically in simulation studies and an analysis of critical care data. Our development also sheds broader light onto the construction of orthogonal learners for other estimands.

Multivariate normal (MVN) probabilities arise in myriad applications, but they are analytically intractable and need to be evaluated via Monte-Carlo-based numerical integration. For the state-of-the-art minimax exponential tilting (MET) method, we show that the complexity of each of its components can be greatly reduced through an integrand parameterization that utilizes the sparse inverse Cholesky factor produced by the Vecchia approximation, whose approximation error is often negligible relative to the Monte-Carlo error. Based on this idea, we derive algorithms that can estimate MVN probabilities and sample from truncated MVN distributions in linear time (and that are easily parallelizable) at the same convergence or acceptance rate as MET, whose complexity is cubic in the dimension of the MVN probability. We showcase the advantages of our methods relative to existing approaches using several simulated examples. We also analyze a groundwater-contamination dataset with over twenty thousand censored measurements to demonstrate the scalability of our method for partially censored Gaussian-process models.

In many applications, a stochastic system is studied using a model implicitly defined via a simulator. We develop a simulation-based parameter inference method for implicitly defined models. Our method differs from traditional likelihood-based inference in that it uses a metamodel for the distribution of a log-likelihood estimator. The metamodel is built on a local asymptotic normality (LAN) property satisfied by the simulation-based log-likelihood estimator under certain conditions. A method for hypothesis test is developed under the metamodel. Our method can enable accurate parameter estimation and uncertainty quantification where other Monte Carlo methods for parameter inference become highly inefficient due to large Monte Carlo variance. We demonstrate our method using numerical examples including a mechanistic model for the population dynamics of infectious disease.

Over the last decades, many prognostic models based on artificial intelligence techniques have been used to provide detailed predictions in healthcare. Unfortunately, the real-world observational data used to train and validate these models are almost always affected by biases that can strongly impact the outcomes validity: two examples are values missing not-at-random and selection bias. Addressing them is a key element in achieving transportability and in studying the causal relationships that are critical in clinical decision making, going beyond simpler statistical approaches based on probabilistic association. In this context, we propose a novel approach that combines selection diagrams, missingness graphs, causal discovery and prior knowledge into a single graphical model to estimate the cardiovascular risk of adolescent and young females who survived breast cancer. We learn this model from data comprising two different cohorts of patients. The resulting causal network model is validated by expert clinicians in terms of risk assessment, accuracy and explainability, and provides a prognostic model that outperforms competing machine learning methods.

Few studies have investigated the diagnostic utilities of biomarkers for predicting bacteremia among septic patients admitted to intensive care units (ICU). Therefore, this study evaluated the prediction power of laboratory biomarkers to utilize those markers with high performance to optimize the predictive model for bacteremia. This retrospective cross-sectional study was conducted at the ICU department of Gyeongsang National University Changwon Hospital in 2019. Adult patients qualifying SEPSIS-3 (increase in sequential organ failure score greater than or equal to 2) criteria with at least two sets of blood culture were selected. Collected data was initially analyzed independently to identify the significant predictors, which was then used to build the multivariable logistic regression (MLR) model. A total of 218 patients with 48 cases of true bacteremia were analyzed in this research. Both CRP and PCT showed a substantial area under the curve (AUC) value for discriminating bacteremia among septic patients (0.757 and 0.845, respectively). To further enhance the predictive accuracy, we combined PCT, bilirubin, neutrophil lymphocyte ratio (NLR), platelets, lactic acid, erythrocyte sedimentation rate (ESR), and Glasgow Coma Scale (GCS) score to build the predictive model with an AUC of 0.907 (95% CI, 0.843 to 0.956). In addition, a high association between bacteremia and mortality rate was discovered through the survival analysis (0.004). While PCT is certainly a useful index for distinguishing patients with and without bacteremia by itself, our MLR model indicates that the accuracy of bacteremia prediction substantially improves by the combined use of PCT, bilirubin, NLR, platelets, lactic acid, ESR, and GCS score.

Uplift modeling is a fundamental component of marketing effect modeling, which is commonly employed to evaluate the effects of treatments on outcomes. Through uplift modeling, we can identify the treatment with the greatest benefit. On the other side, we can identify clients who are likely to make favorable decisions in response to a certain treatment. In the past, uplift modeling approaches relied heavily on the difference-in-difference (DID) architecture, paired with a machine learning model as the estimation learner, while neglecting the link and confidential information between features. We proposed a framework based on graph neural networks that combine causal knowledge with an estimate of uplift value. Firstly, we presented a causal representation technique based on CATE (conditional average treatment effect) estimation and adjacency matrix structure learning. Secondly, we suggested a more scalable uplift modeling framework based on graph convolution networks for combining causal knowledge. Our findings demonstrate that this method works effectively for predicting uplift values, with small errors in typical simulated data, and its effectiveness has been verified in actual industry marketing data.

We study mean-field variational inference in a Bayesian linear model when the sample size n is comparable to the dimension p. In high dimensions, the common approach of minimizing a Kullback-Leibler divergence from the posterior distribution, or maximizing an evidence lower bound, may deviate from the true posterior mean and underestimate posterior uncertainty. We study instead minimization of the TAP free energy, showing in a high-dimensional asymptotic framework that it has a local minimizer which provides a consistent estimate of the posterior marginals and may be used for correctly calibrated posterior inference. Geometrically, we show that the landscape of the TAP free energy is strongly convex in an extensive neighborhood of this local minimizer, which under certain general conditions can be found by an Approximate Message Passing (AMP) algorithm. We then exhibit an efficient algorithm that linearly converges to the minimizer within this local neighborhood. In settings where it is conjectured that no efficient algorithm can find this local neighborhood, we prove analogous geometric properties for a local minimizer of the TAP free energy reachable by AMP, and show that posterior inference based on this minimizer remains correctly calibrated.

We propose a new method for the simultaneous selection and estimation of multivariate sparse additive models with correlated errors. Our method called Covariance Assisted Multivariate Penalized Additive Regression (CoMPAdRe) simultaneously selects among null, linear, and smooth non-linear effects for each predictor while incorporating joint estimation of the sparse residual structure among responses, with the motivation that accounting for inter-response correlation structure can lead to improved accuracy in variable selection and estimation efficiency. CoMPAdRe is constructed in a computationally efficient way that allows the selection and estimation of linear and non-linear covariates to be conducted in parallel across responses. Compared to single-response approaches that marginally select linear and non-linear covariate effects, we demonstrate in simulation studies that the joint multivariate modeling leads to gains in both estimation efficiency and selection accuracy, of greater magnitude in settings where signal is moderate relative to the level of noise. We apply our approach to protein-mRNA expression levels from multiple breast cancer pathways obtained from The Cancer Proteome Atlas and characterize both mRNA-protein associations and protein-protein subnetworks for each pathway. We find non-linear mRNA-protein associations for the Core Reactive, EMT, PIK-AKT, and RTK pathways.

Consider a semi-supervised setting with a labeled dataset of binary responses and predictors and an unlabeled dataset with only the predictors. Logistic regression is equivalent to an exponential tilt model in the labeled population. For semi-supervised estimation, we develop further analysis and understanding of a statistical approach using exponential tilt mixture (ETM) models and maximum nonparametric likelihood estimation, while allowing that the class proportions may differ between the unlabeled and labeled data. We derive asymptotic properties of ETM-based estimation and demonstrate improved efficiency over supervised logistic regression in a random sampling setup and an outcome-stratified sampling setup previously used. Moreover, we reconcile such efficiency improvement with the existing semiparametric efficiency theory when the class proportions in the unlabeled and labeled data are restricted to be the same. We also provide a simulation study to numerically illustrate our theoretical findings.

We study inference on the long-term causal effect of a continual exposure to a novel intervention, which we term a long-term treatment, based on an experiment involving only short-term observations. Key examples include the long-term health effects of regularly-taken medicine or of environmental hazards and the long-term effects on users of changes to an online platform. This stands in contrast to short-term treatments or "shocks," whose long-term effect can reasonably be mediated by short-term observations, enabling the use of surrogate methods. Long-term treatments by definition have direct effects on long-term outcomes via continual exposure so surrogacy cannot reasonably hold. Our approach instead learns long-term temporal dynamics directly from short-term experimental data, assuming that the initial dynamics observed persist but avoiding the need for both surrogacy assumptions and auxiliary data with long-term observations. We connect the problem with offline reinforcement learning, leveraging doubly-robust estimators to estimate long-term causal effects for long-term treatments and construct confidence intervals. Finally, we demonstrate the method in simulated experiments.

This paper aims at building the theoretical foundations for manifold learning algorithms in the space of absolutely continuous probability measures on a compact and convex subset of $\mathbb{R}^d$, metrized with the Wasserstein-2 distance $W$. We begin by introducing a natural construction of submanifolds $Λ$ of probability measures equipped with metric $W_Λ$, the geodesic restriction of $W$ to $Λ$. In contrast to other constructions, these submanifolds are not necessarily flat, but still allow for local linearizations in a similar fashion to Riemannian submanifolds of $\mathbb{R}^d$. We then show how the latent manifold structure of $(Λ,W_Λ)$ can be learned from samples $\{λ_i\}_{i=1}^N$ of $Λ$ and pairwise extrinsic Wasserstein distances $W$ only. In particular, we show that the metric space $(Λ,W_Λ)$ can be asymptotically recovered in the sense of Gromov--Wasserstein from a graph with nodes $\{λ_i\}_{i=1}^N$ and edge weights $W(λ_i,λ_j)$. In addition, we demonstrate how the tangent space at a sample $λ$ can be asymptotically recovered via spectral analysis of a suitable "covariance operator" using optimal transport maps from $λ$ to sufficiently close and diverse samples $\{λ_i\}_{i=1}^N$. The paper closes with some explicit constructions of submanifolds $Λ$ and numerical examples on the recovery of tangent spaces through spectral analysis.

Pairwise measures of dependence are a common tool to map data in the early stages of analysis with several modern examples based on maximized partitions of the pairwise sample space. Following a short survey of modern measures of dependence, we introduce a new measure which recursively splits the ranks of a pair of variables to partition the sample space and computes the $χ^2$ statistic on the resulting bins. Splitting logic is detailed for splits maximizing a score function and randomly selected splits. Simulations indicate that random splitting produces a statistic conservatively approximated by the $χ^2$ distribution without a loss of power to detect numerous different data patterns compared to maximized binning. Though it seems to add no power to detect dependence, maximized recursive binning is shown to produce a natural visualization of the data and the measure. Applying maximized recursive rank binning to S&P 500 constituent data suggests the automatic detection of tail dependence.

Dynamic Item Response Models extend the standard Item Response Theory (IRT) to capture temporal dynamics in learner ability. While these models have the potential to allow instructional systems to actively monitor the evolution of learner proficiency in real time, existing dynamic item response models rely on expensive inference algorithms that scale poorly to massive datasets. In this work, we propose Variational Temporal IRT (VTIRT) for fast and accurate inference of dynamic learner proficiency. VTIRT offers orders of magnitude speedup in inference runtime while still providing accurate inference. Moreover, the proposed algorithm is intrinsically interpretable by virtue of its modular design. When applied to 9 real student datasets, VTIRT consistently yields improvements in predicting future learner performance over other learner proficiency models.