Given the prevalence of missing data in modern statistical research, a broad range of methods is available for any given imputation task. How does one choose the `best' imputation method in a given application? The standard approach is to select some observations, set their status to missing, and compare prediction accuracy of the methods under consideration of these observations. Besides having to somewhat artificially mask observations, a shortcoming of this approach is that imputations based on the conditional mean will rank highest if predictive accuracy is measured with quadratic loss. In contrast, we want to rank highest an imputation that can sample from the true conditional distributions. In this paper, we develop a framework called "Imputation Scores" (I-Scores) for assessing missing value imputations. We provide a specific I-Score based on density ratios and projections, that is applicable to discrete and continuous data. It does not require to mask additional observations for evaluations and is also applicable if there are no complete observations. The population version is shown to be proper in the sense that the highest rank is assigned to an imputation method that samples from the correct conditional distribution. The propriety is shown under the missing completely at random (MCAR) assumption but is also shown to be valid under missing at random (MAR) with slightly more restrictive assumptions. We show empirically on a range of data sets and imputation methods that our score consistently ranks true data high(est) and is able to avoid pitfalls usually associated with performance measures such as RMSE. Finally, we provide the R-package Iscores available on CRAN with an implementation of our method.
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We introduce a computational efficient data-driven framework suitable for quantifying the uncertainty in physical parameters and model formulation of computer models, represented by differential equations. We construct physics-informed priors, which are multi-output GP priors that encode the model's structure in the covariance function. This is extended into a fully Bayesian framework that quantifies the uncertainty of physical parameters and model predictions. Since physical models often are imperfect descriptions of the real process, we allow the model to deviate from the observed data by considering a discrepancy function. For inference, Hamiltonian Monte Carlo is used. Further, approximations for big data are developed that reduce the computational complexity from $\mathcal{O}(N^3)$ to $\mathcal{O}(N\cdot m^2),$ where $m \ll N.$ Our approach is demonstrated in simulation and real data case studies where the physics are described by time-dependent ODEs describe (cardiovascular models) and space-time dependent PDEs (heat equation). In the studies, it is shown that our modelling framework can recover the true parameters of the physical models in cases where 1) the reality is more complex than our modelling choice and 2) the data acquisition process is biased while also producing accurate predictions. Furthermore, it is demonstrated that our approach is computationally faster than traditional Bayesian calibration methods.
Monte Carlo (MC) methods are the most widely used methods to estimate the performance of a policy. Given an interested policy, MC methods give estimates by repeatedly running this policy to collect samples and taking the average of the outcomes. Samples collected during this process are called online samples. To get an accurate estimate, MC methods consume massive online samples. When online samples are expensive, e.g., online recommendations and inventory management, we want to reduce the number of online samples while achieving the same estimate accuracy. To this end, we use off-policy MC methods that evaluate the interested policy by running a different policy called behavior policy. We design a tailored behavior policy such that the variance of the off-policy MC estimator is provably smaller than the ordinary MC estimator. Importantly, this tailored behavior policy can be efficiently learned from existing offline data, i,e., previously logged data, which are much cheaper than online samples. With reduced variance, our off-policy MC method requires fewer online samples to evaluate the performance of a policy compared with the ordinary MC method. Moreover, our off-policy MC estimator is always unbiased.
In this paper we present a novel method, $\textit{Knowledge Persistence}$ ($\mathcal{KP}$), for faster evaluation of Knowledge Graph (KG) completion approaches. Current ranking-based evaluation is quadratic in the size of the KG, leading to long evaluation times and consequently a high carbon footprint. $\mathcal{KP}$ addresses this by representing the topology of the KG completion methods through the lens of topological data analysis, concretely using persistent homology. The characteristics of persistent homology allow $\mathcal{KP}$ to evaluate the quality of the KG completion looking only at a fraction of the data. Experimental results on standard datasets show that the proposed metric is highly correlated with ranking metrics (Hits@N, MR, MRR). Performance evaluation shows that $\mathcal{KP}$ is computationally efficient: In some cases, the evaluation time (validation+test) of a KG completion method has been reduced from 18 hours (using Hits@10) to 27 seconds (using $\mathcal{KP}$), and on average (across methods & data) reduces the evaluation time (validation+test) by $\approx$ $\textbf{99.96}\%$.
Model-based methods have recently shown great potential for off-policy evaluation (OPE); offline trajectories induced by behavioral policies are fitted to transitions of Markov decision processes (MDPs), which are used to rollout simulated trajectories and estimate the performance of policies. Model-based OPE methods face two key challenges. First, as offline trajectories are usually fixed, they tend to cover limited state and action space. Second, the performance of model-based methods can be sensitive to the initialization of their parameters. In this work, we propose the variational latent branching model (VLBM) to learn the transition function of MDPs by formulating the environmental dynamics as a compact latent space, from which the next states and rewards are then sampled. Specifically, VLBM leverages and extends the variational inference framework with the recurrent state alignment (RSA), which is designed to capture as much information underlying the limited training data, by smoothing out the information flow between the variational (encoding) and generative (decoding) part of VLBM. Moreover, we also introduce the branching architecture to improve the model's robustness against randomly initialized model weights. The effectiveness of the VLBM is evaluated on the deep OPE (DOPE) benchmark, from which the training trajectories are designed to result in varied coverage of the state-action space. We show that the VLBM outperforms existing state-of-the-art OPE methods in general.
Verification of discrete time or continuous time dynamical systems over the reals is known to be undecidable. It is however known that undecidability does not hold for various classes of systems: if robustness is defined as the fact that reachability relation is stable under infinitesimal perturbation, then their reachability relation is decidable. In other words, undecidability implies sensitivity under infinitesimal perturbation, a property usually not expected in systems considered in practice, and hence can be seen (somehow informally) as an artefact of the theory, that always assumes exactness. In a similar vein, it is known that, while undecidability holds for logical formulas over the reals, it does not hold when considering delta-undecidability: one must determine whether a property is true, or $\delta$-far from being true. We first extend the previous statements to a theory for general (discrete time, continuous-time, and even hybrid) dynamical systems, and we relate the two approaches. We also relate robustness to some geometric properties of reachability relation. But mainly, when a system is robust, it then makes sense to quantify at which level of perturbation. We prove that assuming robustness to polynomial perturbations on precision leads to reachability verifiable in complexity class PSPACE, and even to a characterization of this complexity class. We prove that assuming robustness to polynomial perturbations on time or length of trajectories leads to similar statements, but with PTIME. It has been recently unexpectedly shown that the length of a solution of a polynomial ordinary differential equation corresponds to a time of computation: PTIME corresponds to solutions of polynomial differential equations of polynomial length. Our results argue that the answer is given by precision: space corresponds to the involved precision.
We present a data structure to randomly sample rows from the Khatri-Rao product of several matrices according to the exact distribution of its leverage scores. Our proposed sampler draws each row in time logarithmic in the height of the Khatri-Rao product and quadratic in its column count, with persistent space overhead at most the size of the input matrices. As a result, it tractably draws samples even when the matrices forming the Khatri-Rao product have tens of millions of rows each. When used to sketch the linear least-squares problems arising in Candecomp / PARAFAC decomposition, our method achieves lower asymptotic complexity per solve than recent state-of-the-art methods. Experiments on billion-scale sparse tensors and synthetic data validate our theoretical claims, with our algorithm achieving higher accuracy than competing methods as the decomposition rank grows.
This paper builds bridges between two families of probabilistic algorithms: (hierarchical) variational inference (VI), which is typically used to model distributions over continuous spaces, and generative flow networks (GFlowNets), which have been used for distributions over discrete structures such as graphs. We demonstrate that, in certain cases, VI algorithms are equivalent to special cases of GFlowNets in the sense of equality of expected gradients of their learning objectives. We then point out the differences between the two families and show how these differences emerge experimentally. Notably, GFlowNets, which borrow ideas from reinforcement learning, are more amenable than VI to off-policy training without the cost of high gradient variance induced by importance sampling. We argue that this property of GFlowNets can provide advantages for capturing diversity in multimodal target distributions.
Stacking (or stacked generalization) is an ensemble learning method with one main distinctiveness from the rest: even though several base models are trained on the original data set, their predictions are further used as input data for one or more metamodels arranged in at least one extra layer. Composing a stack of models can produce high-performance outcomes, but it usually involves a trial-and-error process. Therefore, our previously developed visual analytics system, StackGenVis, was mainly designed to assist users in choosing a set of top-performing and diverse models by measuring their predictive performance. However, it only employs a single logistic regression metamodel. In this paper, we investigate the impact of alternative metamodels on the performance of stacking ensembles using a novel visualization tool, called MetaStackVis. Our interactive tool helps users to visually explore different singular and pairs of metamodels according to their predictive probabilities and multiple validation metrics, as well as their ability to predict specific problematic data instances. MetaStackVis was evaluated with a usage scenario based on a medical data set and via expert interviews.
G-formula is a popular approach for estimating treatment or exposure effects from longitudinal data that are subject to time-varying confounding. G-formula estimation is typically performed by Monte-Carlo simulation, with non-parametric bootstrapping used for inference. We show that G-formula can be implemented by exploiting existing methods for multiple imputation (MI) for synthetic data. This involves using an existing modified version of Rubin's variance estimator. In practice missing data is ubiquitous in longitudinal datasets. We show that such missing data can be readily accommodated as part of the MI procedure, and describe how MI software can be used to implement the approach. We explore its performance using a simulation study.
Verifiability is one of the core editing principles in Wikipedia, where editors are encouraged to provide citations for the added statements. Statements can be any arbitrary piece of text, ranging from a sentence up to a paragraph. However, in many cases, citations are either outdated, missing, or link to non-existing references (e.g. dead URL, moved content etc.). In total, 20\% of the cases such citations refer to news articles and represent the second most cited source. Even in cases where citations are provided, there are no explicit indicators for the span of a citation for a given piece of text. In addition to issues related with the verifiability principle, many Wikipedia entity pages are incomplete, with relevant information that is already available in online news sources missing. Even for the already existing citations, there is often a delay between the news publication time and the reference time. In this thesis, we address the aforementioned issues and propose automated approaches that enforce the verifiability principle in Wikipedia, and suggest relevant and missing news references for further enriching Wikipedia entity pages.
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