Significant recent progress has been made on deriving combination rules that can take as input a set of arbitrarily dependent p-values, and produce as output a single valid p-value. Here, we show that under the assumption of exchangeability of the p-values, many of those rules can be improved (made more powerful). While this observation by itself has practical implications (for example, under repeated tests involving data splitting), it also has implications for combining arbitrarily dependent p-values, since the latter can be made exchangeable by applying a uniformly random permutation. In particular, we derive several simple randomized combination rules for arbitrarily dependent p-values that are more powerful than their deterministic counterparts. For example, we derive randomized and exchangeable improvements of well known p-value combination rules like "twice the median" and "twice the average", as well as geometric and harmonic means. The main technical advance is to show that all these combination rules can be obtained by calibrating the p-values to e-values (using an $\alpha$-dependent calibrator), averaging those e-values, converting to a level $\alpha$ test using Markov's inequality, and finally obtaining p-values by combining this family of tests. The improvements are delivered via recent randomized and exchangeable variants of Markov's inequality.
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Quantifying uncertainty in detected changepoints is an important problem. However it is challenging as the naive approach would use the data twice, first to detect the changes, and then to test them. This will bias the test, and can lead to anti-conservative p-values. One approach to avoid this is to use ideas from post-selection inference, which conditions on the information in the data used to choose which changes to test. As a result this produces valid p-values; that is, p-values that have a uniform distribution if there is no change. Currently such methods have been developed for detecting changes in mean only. This paper presents two approaches for constructing post-selection p-values for detecting changes in variance. These vary depending on the method use to detect the changes, but are general in terms of being applicable for a range of change-detection methods and a range of hypotheses that we may wish to test.
We present a novel method for training score-based generative models which uses nonlinear noising dynamics to improve learning of structured distributions. Generalizing to a nonlinear drift allows for additional structure to be incorporated into the dynamics, thus making the training better adapted to the data, e.g., in the case of multimodality or (approximate) symmetries. Such structure can be obtained from the data by an inexpensive preprocessing step. The nonlinear dynamics introduces new challenges into training which we address in two ways: 1) we develop a new nonlinear denoising score matching (NDSM) method, 2) we introduce neural control variates in order to reduce the variance of the NDSM training objective. We demonstrate the effectiveness of this method on several examples: a) a collection of low-dimensional examples, motivated by clustering in latent space, b) high-dimensional images, addressing issues with mode collapse, small training sets, and approximate symmetries, the latter being a challenge for methods based on equivariant neural networks, which require exact symmetries.
Approximating invariant subspaces of generalized eigenvalue problems (GEPs) is a fundamental computational problem at the core of machine learning and scientific computing. It is, for example, the root of Principal Component Analysis (PCA) for dimensionality reduction, data visualization, and noise filtering, and of Density Functional Theory (DFT), arguably the most popular method to calculate the electronic structure of materials. For a Hermitian definite GEP $HC=SC\Lambda$, let $\Pi_k$ be the true spectral projector on the invariant subspace that is associated with the $k$ smallest (or largest) eigenvalues. Given $H,$ $S$, an integer $k$, and accuracy $\varepsilon\in(0,1)$, we show that we can compute a matrix $\widetilde\Pi_k$ such that $\lVert\Pi_k-\widetilde\Pi_k\rVert_2\leq \varepsilon$, in $O\left( n^{\omega+\eta}\mathrm{polylog}(n,\varepsilon^{-1},\kappa(S),\mathrm{gap}_k^{-1}) \right)$ bit operations in the floating point model with probability $1-1/n$. Here, $\eta>0$ is arbitrarily small, $\omega\lesssim 2.372$ is the matrix multiplication exponent, $\kappa(S)=\lVert S\rVert_2\lVert S^{-1}\rVert_2$, and $\mathrm{gap}_k$ is the gap between eigenvalues $k$ and $k+1$. To the best of our knowledge, this is the first end-to-end analysis achieving such "forward-error" approximation guarantees with nearly $O(n^{\omega+\eta})$ bit complexity, improving classical $\widetilde O(n^3)$ eigensolvers, even for the regular case $(S=I)$. Our methods rely on a new $O(n^{\omega+\eta})$ stability analysis for the Cholesky factorization, and a new smoothed analysis for computing spectral gaps, which can be of independent interest. Ultimately, we obtain new matrix multiplication-type bit complexity upper bounds for PCA problems, including classical PCA and (randomized) low-rank approximation.
In this paper, we develop a new and effective approach to nonparametric quantile regression that accommodates ultrahigh-dimensional data arising from spatio-temporal processes. This approach proves advantageous in staving off computational challenges that constitute known hindrances to existing nonparametric quantile regression methods when the number of predictors is much larger than the available sample size. We investigate conditions under which estimation is feasible and of good overall quality and obtain sharp approximations that we employ to devising statistical inference methodology. These include simultaneous confidence intervals and tests of hypotheses, whose asymptotics is borne by a non-trivial functional central limit theorem tailored to martingale differences. Additionally, we provide finite-sample results through various simulations which, accompanied by an illustrative application to real-worldesque data (on electricity demand), offer guarantees on the performance of the proposed methodology.
This paper is concerned with sample size determination methodology for prediction models. We propose combining the individual calculations via a learning-type curve. We suggest two distinct ways of doing so, a deterministic skeleton of a learning curve and a Gaussian process centred upon its deterministic counterpart. We employ several learning algorithms for modelling the primary endpoint and distinct measures for trial efficacy. We find that the performance may vary with the sample size, but borrowing information across sample size universally improves the performance of such calculations. The Gaussian process-based learning curve appears more robust and statistically efficient, while computational efficiency is comparable. We suggest that anchoring against historical evidence when extrapolating sample sizes should be adopted when such data are available. The methods are illustrated on binary and survival endpoints.
We present a novel formal system for proving quantitative-leakage properties of programs. Based on a theory of Quantitative Information Flow (QIF) that models information leakage as a noisy communication channel, it uses "gain-functions" for the description and measurement of expected leaks. We use a small imperative programming language, augmented with leakage features, and with it express adversaries' activities in the style of, but more generally than, the Hoare triples or expectation transformers that traditionally express deterministic or probabilistic correctness but without information flow. The programs are annotated with "gain-expressions" that capture simple adversarial settings such as "Guess the secret in one try." but also much more general ones; and our formal syntax and logic -based framework enables us to transform such gain-expressions that apply after a program has finished to ones that equivalently apply before the program has begun. In that way we enable a formal proof-based reasoning system for QIF at the source level. We apply it to the %programming language we have chosen, and demonstrate its effectiveness in a number of small but sometimes intricate situations.
This work presents several new results concerning the analysis of the convergence of binary, univariate, and linear subdivision schemes, all related to the {\it contractivity factor} of a convergent scheme. First, we prove that a convergent scheme cannot have a contractivity factor lower than half. Since the lower this factor is, the faster is the convergence of the scheme, schemes with contractivity factor $\frac{1}{2}$, such as those generating spline functions, have optimal convergence rate. Additionally, we provide further insights and conditions for the convergence of linear schemes and demonstrate their applicability in an improved algorithm for determining the convergence of such subdivision schemes.
Similar to the notion of h-adaptivity, where the discretization resolution is adaptively changed, I propose the notion of model adaptivity, where the underlying model (the governing equations) is adaptively changed in space and time. Specifically, this work introduces a hybrid and adaptive coupling of a 3D bulk fluid flow model with a 2D thin film flow model. As a result, this work extends the applicability of existing thin film flow models to complex scenarios where, for example, bulk flow develops into thin films after striking a surface. At each location in space and time, the proposed framework automatically decides whether a 3D model or a 2D model must be applied. Using a meshless approach for both 3D and 2D models, at each particle, the decision to apply a 2D or 3D model is based on the user-prescribed resolution and a local principal component analysis. When a particle needs to be changed from a 3D model to 2D, or vice versa, the discretization is changed, and all relevant data mapping is done on-the-fly. Appropriate two-way coupling conditions and mass conservation considerations between the 3D and 2D models are also developed. Numerical results show that this model adaptive framework shows higher flexibility and compares well against finely resolved 3D simulations. In an actual application scenario, a 3 factor speed up is obtained, while maintaining the accuracy of the solution.
The goal of explainable Artificial Intelligence (XAI) is to generate human-interpretable explanations, but there are no computationally precise theories of how humans interpret AI generated explanations. The lack of theory means that validation of XAI must be done empirically, on a case-by-case basis, which prevents systematic theory-building in XAI. We propose a psychological theory of how humans draw conclusions from saliency maps, the most common form of XAI explanation, which for the first time allows for precise prediction of explainee inference conditioned on explanation. Our theory posits that absent explanation humans expect the AI to make similar decisions to themselves, and that they interpret an explanation by comparison to the explanations they themselves would give. Comparison is formalized via Shepard's universal law of generalization in a similarity space, a classic theory from cognitive science. A pre-registered user study on AI image classifications with saliency map explanations demonstrate that our theory quantitatively matches participants' predictions of the AI.
In recent years, object detection has experienced impressive progress. Despite these improvements, there is still a significant gap in the performance between the detection of small and large objects. We analyze the current state-of-the-art model, Mask-RCNN, on a challenging dataset, MS COCO. We show that the overlap between small ground-truth objects and the predicted anchors is much lower than the expected IoU threshold. We conjecture this is due to two factors; (1) only a few images are containing small objects, and (2) small objects do not appear enough even within each image containing them. We thus propose to oversample those images with small objects and augment each of those images by copy-pasting small objects many times. It allows us to trade off the quality of the detector on large objects with that on small objects. We evaluate different pasting augmentation strategies, and ultimately, we achieve 9.7\% relative improvement on the instance segmentation and 7.1\% on the object detection of small objects, compared to the current state of the art method on MS COCO.
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