Probabilistic linear discriminant analysis (PLDA) has been widely used in open-set verification tasks, such as speaker verification. A potential issue of this model is that the training set often contains limited number of classes, which makes the estimation for the between-class variance unreliable. This unreliable estimation often leads to degraded generalization. In this paper, we present an MAP estimation for the between-class variance, by employing an Inverse-Wishart prior. A key problem is that with hierarchical models such as PLDA, the prior is placed on the variance of class means while the likelihood is based on class members, which makes the posterior inference intractable. We derive a simple MAP estimation for such a model, and test it in both PLDA scoring and length normalization. In both cases, the MAP-based estimation delivers interesting performance improvement.
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Directed Acyclic Graphs (DAGs) provide a powerful framework to model causal relationships among variables in multivariate settings; in addition, through the do-calculus theory, they allow for the identification and estimation of causal effects between variables also from pure observational data. In this setting, the process of inferring the DAG structure from the data is referred to as causal structure learning or causal discovery. We introduce BCDAG, an R package for Bayesian causal discovery and causal effect estimation from Gaussian observational data, implementing the Markov chain Monte Carlo (MCMC) scheme proposed by Castelletti & Mascaro (2021). Our implementation scales efficiently with the number of observations and, whenever the DAGs are sufficiently sparse, with the number of variables in the dataset. The package also provides functions for convergence diagnostics and for visualizing and summarizing posterior inference. In this paper, we present the key features of the underlying methodology along with its implementation in BCDAG. We then illustrate the main functions and algorithms on both real and simulated datasets.
Verification of probabilistic forecasts for extreme events has been a very active field of research, stirred by media and public opinions who naturally focus their attention on extreme events, and easily draw biased onclusions. In this context, classical verification methodologies tailored for extreme events, such as thresholded and weighted scoring rules, have undesirable properties that cannot be mitigated; the well-known Continuous Ranked Probability Score (CRPS) makes no exception. In this paper, we define a formal framework to assess the behavior of forecast evaluation procedures with respect to extreme events, that we use to point out that assessment based on the expectation of a proper score is not suitable for extremes. As an alternative, we propose to study the properties of the CRPS as a random variable using extreme value theory to address extreme events verification. To compare calibrated forecasts, an index is introduced that summarizes the ability of probabilistic forecasts to predict extremes. Its strengths and limitations are discussed using both theoretical arguments and simulations.
In this paper we propose a new optimization model for maximum likelihood estimation of causal and invertible ARMA models. Through a set of numerical experiments we show how our proposed model outperforms, both in terms of quality of the fitted model as well as in the computational time, the classical estimation procedure based on Jones reparametrization. We also propose a regularization term in the model and we show how this addition improves the out of sample quality of the fitted model. This improvement is achieved thanks to an increased penalty on models close to the non causality or non invertibility boundary.
We consider the task of estimating the causal effect of a treatment variable on a long-term outcome variable using data from an observational domain and an experimental domain. The observational data is assumed to be confounded and hence without further assumptions, this dataset alone cannot be used for causal inference. Also, only a short-term version of the primary outcome variable of interest is observed in the experimental data, and hence, this dataset alone cannot be used for causal inference either. In a recent work, Athey et al. (2020) proposed a method for systematically combining such data for identifying the downstream causal effect in view. Their approach is based on the assumptions of internal and external validity of the experimental data, and an extra novel assumption called latent unconfoundedness. In this paper, we first review their proposed approach and discuss the latent unconfoundedness assumption. Then we propose two alternative approaches for data fusion for the purpose of estimating average treatment effect as well as the effect of treatment on the treated. Our first proposed approach is based on assuming equi-confounding bias for the short-term and long-term outcomes. Our second proposed approach is based on the proximal causal inference framework, in which we assume the existence of an extra variable in the system which is a proxy of the latent confounder of the treatment-outcome relation.
In network analysis, how to estimate the number of communities $K$ is a fundamental problem. We consider a broad setting where we allow severe degree heterogeneity and a wide range of sparsity levels, and propose Stepwise Goodness-of-Fit (StGoF) as a new approach. This is a stepwise algorithm, where for $m = 1, 2, \ldots$, we alternately use a community detection step and a goodness-of-fit (GoF) step. We adapt SCORE \cite{SCORE} for community detection, and propose a new GoF metric. We show that at step $m$, the GoF metric diverges to $\infty$ in probability for all $m < K$ and converges to $N(0,1)$ if $m = K$. This gives rise to a consistent estimate for $K$. Also, we discover the right way to define the signal-to-noise ratio (SNR) for our problem and show that consistent estimates for $K$ do not exist if $\mathrm{SNR} \goto 0$, and StGoF is uniformly consistent for $K$ if $\mathrm{SNR} \goto \infty$. Therefore, StGoF achieves the optimal phase transition. Similar stepwise methods (e.g., \cite{wang2017likelihood, ma2018determining}) are known to face analytical challenges. We overcome the challenges by using a different stepwise scheme in StGoF and by deriving sharp results that are not available before. The key to our analysis is to show that SCORE has the {\it Non-Splitting Property (NSP)}. Primarily due to a non-tractable rotation of eigenvectors dictated by the Davis-Kahan $\sin(\theta)$ theorem, the NSP is non-trivial to prove and requires new techniques we develop.
Statistical divergences (SDs), which quantify the dissimilarity between probability distributions, are a basic constituent of statistical inference and machine learning. A modern method for estimating those divergences relies on parametrizing an empirical variational form by a neural network (NN) and optimizing over parameter space. Such neural estimators are abundantly used in practice, but corresponding performance guarantees are partial and call for further exploration. We establish non-asymptotic absolute error bounds for a neural estimator realized by a shallow NN, focusing on four popular $\mathsf{f}$-divergences -- Kullback-Leibler, chi-squared, squared Hellinger, and total variation. Our analysis relies on non-asymptotic function approximation theorems and tools from empirical process theory to bound the two sources of error involved: function approximation and empirical estimation. The bounds characterize the effective error in terms of NN size and the number of samples, and reveal scaling rates that ensure consistency. For compactly supported distributions, we further show that neural estimators of the first three divergences above with appropriate NN growth-rate are minimax rate-optimal, achieving the parametric convergence rate.
A fundamental goal of scientific research is to learn about causal relationships. However, despite its critical role in the life and social sciences, causality has not had the same importance in Natural Language Processing (NLP), which has traditionally placed more emphasis on predictive tasks. This distinction is beginning to fade, with an emerging area of interdisciplinary research at the convergence of causal inference and language processing. Still, research on causality in NLP remains scattered across domains without unified definitions, benchmark datasets and clear articulations of the remaining challenges. In this survey, we consolidate research across academic areas and situate it in the broader NLP landscape. We introduce the statistical challenge of estimating causal effects, encompassing settings where text is used as an outcome, treatment, or as a means to address confounding. In addition, we explore potential uses of causal inference to improve the performance, robustness, fairness, and interpretability of NLP models. We thus provide a unified overview of causal inference for the computational linguistics community.
Sampling methods (e.g., node-wise, layer-wise, or subgraph) has become an indispensable strategy to speed up training large-scale Graph Neural Networks (GNNs). However, existing sampling methods are mostly based on the graph structural information and ignore the dynamicity of optimization, which leads to high variance in estimating the stochastic gradients. The high variance issue can be very pronounced in extremely large graphs, where it results in slow convergence and poor generalization. In this paper, we theoretically analyze the variance of sampling methods and show that, due to the composite structure of empirical risk, the variance of any sampling method can be decomposed into \textit{embedding approximation variance} in the forward stage and \textit{stochastic gradient variance} in the backward stage that necessities mitigating both types of variance to obtain faster convergence rate. We propose a decoupled variance reduction strategy that employs (approximate) gradient information to adaptively sample nodes with minimal variance, and explicitly reduces the variance introduced by embedding approximation. We show theoretically and empirically that the proposed method, even with smaller mini-batch sizes, enjoys a faster convergence rate and entails a better generalization compared to the existing methods.
Implicit probabilistic models are models defined naturally in terms of a sampling procedure and often induces a likelihood function that cannot be expressed explicitly. We develop a simple method for estimating parameters in implicit models that does not require knowledge of the form of the likelihood function or any derived quantities, but can be shown to be equivalent to maximizing likelihood under some conditions. Our result holds in the non-asymptotic parametric setting, where both the capacity of the model and the number of data examples are finite. We also demonstrate encouraging experimental results.
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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