Quasi-experimental research designs, such as regression discontinuity and interrupted time series, allow for causal inference in the absence of a randomized controlled trial, at the cost of additional assumptions. In this paper, we provide a framework for discontinuity-based designs using Bayesian model comparison and Gaussian process regression, which we refer to as 'Bayesian nonparametric discontinuity design', or BNDD for short. BNDD addresses the two major shortcomings in most implementations of such designs: overconfidence due to implicit conditioning on the alleged effect, and model misspecification due to reliance on overly simplistic regression models. With the appropriate Gaussian process covariance function, our approach can detect discontinuities of any order, and in spectral features. We demonstrate the usage of BNDD in simulations, and apply the framework to determine the effect of running for political positions on longevity, of the effect of an alleged historical phantom border in the Netherlands on Dutch voting behaviour, and of Kundalini Yoga meditation on heart rate.
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Processing 是一門開源編程語言和與之配套的集成開發環境（IDE）的名稱。Processing 在電子藝術和視覺設計社區被用來教授編程基礎，并運用于大量的新媒體和互動藝術作品中。
Consider the problem of nonparametric estimation of an unknown $\beta$-H\"older smooth density $p_{XY}$ at a given point, where $X$ and $Y$ are both $d$ dimensional. An infinite sequence of i.i.d.\ samples $(X_i,Y_i)$ are generated according to this distribution, and two terminals observe $(X_i)$ and $(Y_i)$, respectively. They are allowed to exchange $k$ bits either in oneway or interactively in order for Bob to estimate the unknown density. We show that the minimax mean square risk is order $\left(\frac{k}{\log k} \right)^{-\frac{2\beta}{d+2\beta}}$ for one-way protocols and $k^{-\frac{2\beta}{d+2\beta}}$ for interactive protocols. The logarithmic improvement is nonexistent in the parametric counterparts, and therefore can be regarded as a consequence of nonparametric nature of the problem. Moreover, a few rounds of interactions achieve the interactive minimax rate: the number of rounds can grow as slowly as the super-logarithm (i.e., inverse tetration) of $k$. The proof of the upper bound is based on a novel multi-round scheme for estimating the joint distribution of a pair of biased Bernoulli variables.
Skills or low-level policies in reinforcement learning are temporally extended actions that can speed up learning and enable complex behaviours. Recent work in offline reinforcement learning and imitation learning has proposed several techniques for skill discovery from a set of expert trajectories. While these methods are promising, the number K of skills to discover is always a fixed hyperparameter, which requires either prior knowledge about the environment or an additional parameter search to tune it. We first propose a method for offline learning of options (a particular skill framework) exploiting advances in variational inference and continuous relaxations. We then highlight an unexplored connection between Bayesian nonparametrics and offline skill discovery, and show how to obtain a nonparametric version of our model. This version is tractable thanks to a carefully structured approximate posterior with a dynamically-changing number of options, removing the need to specify K. We also show how our nonparametric extension can be applied in other skill frameworks, and empirically demonstrate that our method can outperform state-of-the-art offline skill learning algorithms across a variety of environments. Our code is available at //github.com/layer6ai-labs/BNPO .
Based on recent developments in optimal transport theory, we propose a novel model-selection strategy for Bayesian learning. More precisely, the goal of this paper is to introduce the Wasserstein barycenter of the posterior law on models, as a Bayesian predictive posterior, alternative to classical choices such as the maximum a posteriori and the model average Bayesian estimators. After formulating the general problem of Bayesian model selection in a common, parameter-free framework, we exhibit conditions granting the existence and statistical consistency of this estimator, discuss some of its general and specific properties, and provide insight into its theoretical advantages. Furthermore, we illustrate how it can be computed using the theoretical stochastic gradient descent (SGD) algorithm in Wasserstein space introduced in a companion paper arXiv:2201.04232v2 [math.OC] , and provide a numerical example for experimental validation of the proposed method.
Distribution forecast can quantify forecast uncertainty and provide various forecast scenarios with their corresponding estimated probabilities. Accurate distribution forecast is crucial for planning - for example when making production capacity or inventory allocation decisions. We propose a practical and robust distribution forecast framework that relies on backtest-based bootstrap and adaptive residual selection. The proposed approach is robust to the choice of the underlying forecasting model, accounts for uncertainty around the input covariates, and relaxes the independence between residuals and covariates assumption. It reduces the Absolute Coverage Error by more than 63% compared to the classic bootstrap approaches and by 2% - 32% compared to a variety of State-of-the-Art deep learning approaches on in-house product sales data and M4-hourly competition data.
Recently geometric hypergraphs that can be defined by intersections of pseudohalfplanes with a finite point set were defined in a purely combinatorial way. This led to extensions of earlier results about points and halfplanes to pseudohalfplanes, including polychromatic colorings and discrete Helly-type theorems about pseudohalfplanes. Here we continue this line of research and introduce the notion of convex sets of such pseudohalfplane hypergraphs. In this context we prove several results corresponding to classical results about convexity, namely Helly Theorem, Carath\'eodory's Theorem, Kirchberger's Theorem, Separation Theorem, Radon's Theorem and the Cup-Cap Theorem. These results imply the respective results about pseudoconvex sets in the plane defined using pseudohalfplanes. It turns out that most of our results can be also proved using oriented matroids and topological affine planes (TAPs) but our approach is different from both of them. Compared to oriented matroids, our theory is based on a linear ordering of the vertex set which makes our definitions and proofs quite different and perhaps more elementary. Compared to TAPs, which are continuous objects, our proofs are purely combinatorial and again quite different in flavor. Altogether, we believe that our new approach can further our understanding of these fundamental convexity results.
A key advantage of isogeometric discretizations is their accurate and well-behaved eigenfrequencies and eigenmodes. For degree two and higher, however, optical branches of spurious outlier frequencies and modes may appear due to boundaries or reduced continuity at patch interfaces. In this paper, we introduce a variational approach based on perturbed eigenvalue analysis that eliminates outlier frequencies without negatively affecting the accuracy in the remainder of the spectrum and modes. We then propose a pragmatic iterative procedure that estimates the perturbation parameters in such a way that the outlier frequencies are effectively reduced. We demonstrate that our approach allows for a much larger critical time-step size in explicit dynamics calculations. In addition, we show that the critical time-step size obtained with the proposed approach does not depend on the polynomial degree of spline basis functions.
This work focuses on mitigating two limitations in the joint learning of local feature detectors and descriptors. First, the ability to estimate the local shape (scale, orientation, etc.) of feature points is often neglected during dense feature extraction, while the shape-awareness is crucial to acquire stronger geometric invariance. Second, the localization accuracy of detected keypoints is not sufficient to reliably recover camera geometry, which has become the bottleneck in tasks such as 3D reconstruction. In this paper, we present ASLFeat, with three light-weight yet effective modifications to mitigate above issues. First, we resort to deformable convolutional networks to densely estimate and apply local transformation. Second, we take advantage of the inherent feature hierarchy to restore spatial resolution and low-level details for accurate keypoint localization. Finally, we use a peakiness measurement to relate feature responses and derive more indicative detection scores. The effect of each modification is thoroughly studied, and the evaluation is extensively conducted across a variety of practical scenarios. State-of-the-art results are reported that demonstrate the superiority of our methods.
This work focuses on combining nonparametric topic models with Auto-Encoding Variational Bayes (AEVB). Specifically, we first propose iTM-VAE, where the topics are treated as trainable parameters and the document-specific topic proportions are obtained by a stick-breaking construction. The inference of iTM-VAE is modeled by neural networks such that it can be computed in a simple feed-forward manner. We also describe how to introduce a hyper-prior into iTM-VAE so as to model the uncertainty of the prior parameter. Actually, the hyper-prior technique is quite general and we show that it can be applied to other AEVB based models to alleviate the {\it collapse-to-prior} problem elegantly. Moreover, we also propose HiTM-VAE, where the document-specific topic distributions are generated in a hierarchical manner. HiTM-VAE is even more flexible and can generate topic distributions with better variability. Experimental results on 20News and Reuters RCV1-V2 datasets show that the proposed models outperform the state-of-the-art baselines significantly. The advantages of the hyper-prior technique and the hierarchical model construction are also confirmed by experiments.
Discrete random structures are important tools in Bayesian nonparametrics and the resulting models have proven effective in density estimation, clustering, topic modeling and prediction, among others. In this paper, we consider nested processes and study the dependence structures they induce. Dependence ranges between homogeneity, corresponding to full exchangeability, and maximum heterogeneity, corresponding to (unconditional) independence across samples. The popular nested Dirichlet process is shown to degenerate to the fully exchangeable case when there are ties across samples at the observed or latent level. To overcome this drawback, inherent to nesting general discrete random measures, we introduce a novel class of latent nested processes. These are obtained by adding common and group-specific completely random measures and, then, normalising to yield dependent random probability measures. We provide results on the partition distributions induced by latent nested processes, and develop an Markov Chain Monte Carlo sampler for Bayesian inferences. A test for distributional homogeneity across groups is obtained as a by product. The results and their inferential implications are showcased on synthetic and real data.
In this paper, we study the optimal convergence rate for distributed convex optimization problems in networks. We model the communication restrictions imposed by the network as a set of affine constraints and provide optimal complexity bounds for four different setups, namely: the function $F(\xb) \triangleq \sum_{i=1}^{m}f_i(\xb)$ is strongly convex and smooth, either strongly convex or smooth or just convex. Our results show that Nesterov's accelerated gradient descent on the dual problem can be executed in a distributed manner and obtains the same optimal rates as in the centralized version of the problem (up to constant or logarithmic factors) with an additional cost related to the spectral gap of the interaction matrix. Finally, we discuss some extensions to the proposed setup such as proximal friendly functions, time-varying graphs, improvement of the condition numbers.
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