Parameter inference, i.e. inferring the posterior distribution of the parameters of a statistical model given some data, is a central problem to many scientific disciplines. Generative models can be used as an alternative to Markov Chain Monte Carlo methods for conducting posterior inference, both in likelihood-based and simulation-based problems. However, assessing the accuracy of posteriors encoded in generative models is not straightforward. In this paper, we introduce `Tests of Accuracy with Random Points' (TARP) coverage testing as a method to estimate coverage probabilities of generative posterior estimators. Our method differs from previously-existing coverage-based methods, which require posterior evaluations. We prove that our approach is necessary and sufficient to show that a posterior estimator is accurate. We demonstrate the method on a variety of synthetic examples, and show that TARP can be used to test the results of posterior inference analyses in high-dimensional spaces. We also show that our method can detect inaccurate inferences in cases where existing methods fail.
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The big data era of science and technology motivates statistical modeling of matrix-valued data using a low-rank representation that simultaneously summarizes key characteristics of the data and enables dimension reduction for data compression and storage. Low-rank representations such as singular value decomposition factor the original data into the product of orthonormal basis functions and weights, where each basis function represents an independent feature of the data. However, the basis functions in these factorizations are typically computed using algorithmic methods that cannot quantify uncertainty or account for explicit structure beyond what is implicitly specified via data correlation. We propose a flexible prior distribution for orthonormal matrices that can explicitly model structure in the basis functions. The prior is used within a general probabilistic model for singular value decomposition to conduct posterior inference on the basis functions while accounting for measurement error and fixed effects. To contextualize the proposed prior and model, we discuss how the prior specification can be used for various scenarios and relate the model to its deterministic counterpart. We demonstrate favorable model properties through synthetic data examples and apply our method to sea surface temperature data from the northern Pacific, enhancing our understanding of the ocean's internal variability.
The selection of Gaussian kernel parameters plays an important role in the applications of support vector classification (SVC). A commonly used method is the k-fold cross validation with grid search (CV), which is extremely time-consuming because it needs to train a large number of SVC models. In this paper, a new approach is proposed to train SVC and optimize the selection of Gaussian kernel parameters. We first formulate the training and parameter selection of SVC as a minimax optimization problem named as MaxMin-L2-SVC-NCH, in which the minimization problem is an optimization problem of finding the closest points between two normal convex hulls (L2-SVC-NCH) while the maximization problem is an optimization problem of finding the optimal Gaussian kernel parameters. A lower time complexity can be expected in MaxMin-L2-SVC-NCH because CV is not needed. We then propose a projected gradient algorithm (PGA) for training L2-SVC-NCH. The famous sequential minimal optimization (SMO) algorithm is a special case of the PGA. Thus, the PGA can provide more flexibility than the SMO. Furthermore, the solution of the maximization problem is done by a gradient ascent algorithm with dynamic learning rate. The comparative experiments between MaxMin-L2-SVC-NCH and the previous best approaches on public datasets show that MaxMin-L2-SVC-NCH greatly reduces the number of models to be trained while maintaining competitive test accuracy. These findings indicate that MaxMin-L2-SVC-NCH is a better choice for SVC tasks.
Randomized controlled trials (RCTs) are a cornerstone of comparative effectiveness because they remove the confounding bias present in observational studies. However, RCTs are typically much smaller than observational studies because of financial and ethical considerations. Therefore it is of great interest to be able to incorporate plentiful observational data into the analysis of smaller RCTs. Previous estimators developed for this purpose rely on unrealistic additional assumptions without which the added data can bias the effect estimate. Recent work proposed an alternative method (prognostic adjustment) that imposes no additional assumption and increases efficiency in the analysis of RCTs. The idea is to use the observational data to learn a prognostic model: a regression of the outcome onto the covariates. The predictions from this model, generated from the RCT subjects' baseline variables, are used as a covariate in a linear model. In this work, we extend this framework to work when conducting inference with nonparametric efficient estimators in trial analysis. Using simulations, we find that this approach provides greater power (i.e., smaller standard errors) than without prognostic adjustment, especially when the trial is small. We also find that the method is robust to observed or unobserved shifts between the observational and trial populations and does not introduce bias. Lastly, we showcase this estimator leveraging real-world historical data on a randomized blood transfusion study of trauma patients.
In Bayesian probabilistic programming, a central problem is to estimate the normalised posterior distribution (NPD) of a probabilistic program with conditioning. Prominent approximate approaches to address this problem include Markov chain Monte Carlo and variational inference, but neither can generate guaranteed outcomes within limited time. Moreover, most existing formal approaches that perform exact inference for NPD are restricted to programs with closed-form solutions or bounded loops/recursion. A recent work (Beutner et al., PLDI 2022) derived guaranteed bounds for NPD over programs with unbounded recursion. However, as this approach requires recursion unrolling, it suffers from the path explosion problem. Furthermore, previous approaches do not consider score-recursive probabilistic programs that allow score statements inside loops, which is non-trivial and requires careful treatment to ensure the integrability of the normalising constant in NPD. In this work, we propose a novel automated approach to derive bounds for NPD via polynomial templates. Our approach can handle probabilistic programs with unbounded while loops and continuous distributions with infinite supports. The novelties in our approach are three-fold: First, we use polynomial templates to circumvent the path explosion problem from recursion unrolling; Second, we derive a novel multiplicative variant of Optional Stopping Theorem that addresses the integrability issue in score-recursive programs; Third, to increase the accuracy of the derived bounds via polynomial templates, we propose a novel technique of truncation that truncates a program into a bounded range of program values. Experiments over a wide range of benchmarks demonstrate that our approach is time-efficient and can derive bounds for NPD that are comparable with (or tighter than) the recursion-unrolling approach (Beutner et al., PLDI 2022).
Model-based control requires an accurate model of the system dynamics for precisely and safely controlling the robot in complex and dynamic environments. Moreover, in the presence of variations in the operating conditions, the model should be continuously refined to compensate for dynamics changes. In this paper, we present a self-supervised learning approach that actively models the dynamics of nonlinear robotic systems. We combine offline learning from past experience and online learning from current robot interaction with the unknown environment. These two ingredients enable a highly sample-efficient and adaptive learning process, capable of accurately inferring model dynamics in real-time even in operating regimes that greatly differ from the training distribution. Moreover, we design an uncertainty-aware model predictive controller that is heuristically conditioned to the aleatoric (data) uncertainty of the learned dynamics. This controller actively chooses the optimal control actions that (i) optimize the control performance and (ii) improve the efficiency of online learning sample collection. We demonstrate the effectiveness of our method through a series of challenging real-world experiments using a quadrotor system. Our approach showcases high resilience and generalization capabilities by consistently adapting to unseen flight conditions, while it significantly outperforms classical and adaptive control baselines.
Deep Neural Networks often inherit spurious correlations embedded in training data and hence may fail to generalize to unseen domains, which have different distributions from the domain to provide training data. M. Arjovsky et al. (2019) introduced the concept out-of-distribution (o.o.d.) risk, which is the maximum risk among all domains, and formulated the issue caused by spurious correlations as a minimization problem of the o.o.d. risk. Invariant Risk Minimization (IRM) is considered to be a promising approach to minimize the o.o.d. risk: IRM estimates a minimum of the o.o.d. risk by solving a bi-level optimization problem. While IRM has attracted considerable attention with empirical success, it comes with few theoretical guarantees. Especially, a solid theoretical guarantee that the bi-level optimization problem gives the minimum of the o.o.d. risk has not yet been established. Aiming at providing a theoretical justification for IRM, this paper rigorously proves that a solution to the bi-level optimization problem minimizes the o.o.d. risk under certain conditions. The result also provides sufficient conditions on distributions providing training data and on a dimension of feature space for the bi-leveled optimization problem to minimize the o.o.d. risk.
The number of modes in a probability density function is representative of the model's complexity and can also be viewed as the number of existing subpopulations. Despite its relevance, little research has been devoted to its estimation. Focusing on the univariate setting, we propose a novel approach targeting prediction accuracy inspired by some overlooked aspects of the problem. We argue for the need for structure in the solutions, the subjective and uncertain nature of modes, and the convenience of a holistic view blending global and local density properties. Our method builds upon a combination of flexible kernel estimators and parsimonious compositional splines. Feature exploration, model selection and mode testing are implemented in the Bayesian inference paradigm, providing soft solutions and allowing to incorporate expert judgement in the process. The usefulness of our proposal is illustrated through a case study in sports analytics, showcasing multiple companion visualisation tools. A thorough simulation study demonstrates that traditional modality-driven approaches paradoxically struggle to provide accurate results. In this context, our method emerges as a top-tier alternative offering innovative solutions for analysts.
Sampling from diffusion probabilistic models (DPMs) can be viewed as a piecewise distribution transformation, which generally requires hundreds or thousands of steps of the inverse diffusion trajectory to get a high-quality image. Recent progress in designing fast samplers for DPMs achieves a trade-off between sampling speed and sample quality by knowledge distillation or adjusting the variance schedule or the denoising equation. However, it can't be optimal in both aspects and often suffer from mode mixture in short steps. To tackle this problem, we innovatively regard inverse diffusion as an optimal transport (OT) problem between latents at different stages and propose the DPM-OT, a unified learning framework for fast DPMs with a direct expressway represented by OT map, which can generate high-quality samples within around 10 function evaluations. By calculating the semi-discrete optimal transport map between the data latents and the white noise, we obtain an expressway from the prior distribution to the data distribution, while significantly alleviating the problem of mode mixture. In addition, we give the error bound of the proposed method, which theoretically guarantees the stability of the algorithm. Extensive experiments validate the effectiveness and advantages of DPM-OT in terms of speed and quality (FID and mode mixture), thus representing an efficient solution for generative modeling. Source codes are available at //github.com/cognaclee/DPM-OT
The ability to envision future states is crucial to informed decision making while interacting with dynamic environments. With cameras providing a prevalent and information rich sensing modality, the problem of predicting future states from image sequences has garnered a lot of attention. Current state of the art methods typically train large parametric models for their predictions. Though often able to predict with accuracy, these models rely on the availability of large training datasets to converge to useful solutions. In this paper we focus on the problem of predicting future images of an image sequence from very little training data. To approach this problem, we use non-parametric models to take a probabilistic approach to image prediction. We generate probability distributions over sequentially predicted images and propagate uncertainty through time to generate a confidence metric for our predictions. Gaussian Processes are used for their data efficiency and ability to readily incorporate new training data online. We showcase our method by successfully predicting future frames of a smooth fluid simulation environment.
Understanding dynamics in complex systems is challenging because there are many degrees of freedom, and those that are most important for describing events of interest are often not obvious. The leading eigenfunctions of the transition operator are useful for visualization, and they can provide an efficient basis for computing statistics such as the likelihood and average time of events (predictions). Here we develop inexact iterative linear algebra methods for computing these eigenfunctions (spectral estimation) and making predictions from a data set of short trajectories sampled at finite intervals. We demonstrate the methods on a low-dimensional model that facilitates visualization and a high-dimensional model of a biomolecular system. Implications for the prediction problem in reinforcement learning are discussed.
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