Adrien Saumard, Jon A. Wellner.

Source: Statistics Surveys, Volume 8, 45--114.

Abstract:
We review and formulate results concerning log-concavity and strong-log-concavity in both discrete and continuous settings. We show how preservation of log-concavity and strong log-concavity on $mathbb{R}$ under convolution follows from a fundamental monotonicity result of Efron (1965). We provide a new proof of Efron’s theorem using the recent asymmetric Brascamp-Lieb inequality due to Otto and Menz (2013). Along the way we review connections between log-concavity and other areas of mathematics and statistics, including concentration of measure, log-Sobolev inequalities, convex geometry, MCMC algorithms, Laplace approximations, and machine learning.

Bertrand Clarke, Jennifer Clarke

Source: Statist. Surv., Volume 6, 1--73.

Abstract:
We review predictive techniques from several traditional branches of statistics. Starting with prediction based on the normal model and on the empirical distribution function, we proceed to techniques for various forms of regression and classification. Then, we turn to time series, longitudinal data, and survival analysis. Our focus throughout is on the mechanics of prediction more than on the properties of predictors.

Gregory J. Matthews, Ofer Harel

Source: Statist. Surv., Volume 5, 1--29.

Abstract:
There is an ever increasing demand from researchers for access to useful microdata files. However, there are also growing concerns regarding the privacy of the individuals contained in the microdata. Ideally, microdata could be released in such a way that a balance between usefulness of the data and privacy is struck. This paper presents a review of proposed methods of statistical disclosure control and techniques for assessing the privacy of such methods under different definitions of disclosure.

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A. Philip Dawid, Vanessa Didelez

Source: Statist. Surv., Volume 4, 184--231.

Abstract:
We consider the problem of learning about and comparing the consequences of dynamic treatment strategies on the basis of observational data. We formulate this within a probabilistic decision-theoretic framework. Our approach is compared with related work by Robins and others: in particular, we show how Robins’s ‘ G -computation’ algorithm arises naturally from this decision-theoretic perspective. Careful attention is paid to the mathematical and substantive conditions required to justify the use of this formula. These conditions revolve around a property we term stability , which relates the probabilistic behaviours of observational and interventional regimes. We show how an assumption of ‘sequential randomization’ (or ‘no unmeasured confounders’), or an alternative assumption of ‘sequential irrelevance’, can be used to infer stability. Probabilistic influence diagrams are used to simplify manipulations, and their power and limitations are discussed. We compare our approach with alternative formulations based on causal DAGs or potential response models. We aim to show that formulating the problem of assessing dynamic treatment strategies as a problem of decision analysis brings clarity, simplicity and generality.

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Arctic sea ice extent (SIE) in September 2019 ranked second-to-lowest in history and is trending downward. The understanding of how internal variability amplifies the effects of external $ ext{CO}_2$ forcing is still limited. We propose the VARCTIC, which is a Vector Autoregression (VAR) designed to capture and extrapolate Arctic feedback loops. VARs are dynamic simultaneous systems of equations, routinely estimated to predict and understand the interactions of multiple macroeconomic time series. Hence, the VARCTIC is a parsimonious compromise between fullblown climate models and purely statistical approaches that usually offer little explanation of the underlying mechanism. Our "business as usual" completely unconditional forecast has SIE hitting 0 in September by the 2060s. Impulse response functions reveal that anthropogenic $ ext{CO}_2$ emission shocks have a permanent effect on SIE - a property shared by no other shock. Further, we find Albedo- and Thickness-based feedbacks to be the main amplification channels through which $ ext{CO}_2$ anomalies impact SIE in the short/medium run. Conditional forecast analyses reveal that the future path of SIE crucially depends on the evolution of $ ext{CO}_2$ emissions, with outcomes ranging from recovering SIE to it reaching 0 in the 2050s. Finally, Albedo and Thickness feedbacks are shown to play an important role in accelerating the speed at which predicted SIE is heading towards 0.

We show, by an explicit construction, that a mixture of univariate Gaussians with variance 1 and means in $[-A,A]$ can have $Omega(A^2)$ modes. This disproves a recent conjecture of Dytso, Yagli, Poor and Shamai [IEEE Trans. Inform. Theory, Apr. 2020], who showed that such a mixture can have at most $O(A^2)$ modes and surmised that the upper bound could be improved to $O(A)$. Our result holds even if an additional variance constraint is imposed on the mixing distribution. Extending the result to higher dimensions, we exhibit a mixture of Gaussians in $mathbb{R}^d$, with identity covariances and means inside $[-A,A]^d$, that has $Omega(A^{2d})$ modes.

A distributed binary hypothesis testing (HT) problem over a noisy (discrete and memoryless) channel studied previously by the authors is investigated from the perspective of the strong converse property. It was shown by Ahlswede and Csisz'{a}r that a strong converse holds in the above setting when the channel is rate-limited and noiseless. Motivated by this observation, we show that the strong converse continues to hold in the noisy channel setting for a special case of HT known as testing against independence (TAI), under the assumption that the channel transition matrix has non-zero elements. The proof utilizes the blowing up lemma and the recent change of measure technique of Tyagi and Watanabe as the key tools.

A major difficulty of solving continuous POMDPs is to infer the multi-modal distribution of the unobserved true states and to make the planning algorithm dependent on the perceived uncertainty. We cast POMDP filtering and planning problems as two closely related Sequential Monte Carlo (SMC) processes, one over the real states and the other over the future optimal trajectories, and combine the merits of these two parts in a new model named the DualSMC network. In particular, we first introduce an adversarial particle filter that leverages the adversarial relationship between its internal components. Based on the filtering results, we then propose a planning algorithm that extends the previous SMC planning approach [Piche et al., 2018] to continuous POMDPs with an uncertainty-dependent policy. Crucially, not only can DualSMC handle complex observations such as image input but also it remains highly interpretable. It is shown to be effective in three continuous POMDP domains: the floor positioning domain, the 3D light-dark navigation domain, and a modified Reacher domain.

Adaptive importance samplers are adaptive Monte Carlo algorithms to estimate expectations with respect to some target distribution which extit{adapt} themselves to obtain better estimators over a sequence of iterations. Although it is straightforward to show that they have the same $mathcal{O}(1/sqrt{N})$ convergence rate as standard importance samplers, where $N$ is the number of Monte Carlo samples, the behaviour of adaptive importance samplers over the number of iterations has been left relatively unexplored. In this work, we investigate an adaptation strategy based on convex optimisation which leads to a class of adaptive importance samplers termed extit{optimised adaptive importance samplers} (OAIS). These samplers rely on the iterative minimisation of the $chi^2$-divergence between an exponential-family proposal and the target. The analysed algorithms are closely related to the class of adaptive importance samplers which minimise the variance of the weight function. We first prove non-asymptotic error bounds for the mean squared errors (MSEs) of these algorithms, which explicitly depend on the number of iterations and the number of samples together. The non-asymptotic bounds derived in this paper imply that when the target belongs to the exponential family, the $L_2$ errors of the optimised samplers converge to the optimal rate of $mathcal{O}(1/sqrt{N})$ and the rate of convergence in the number of iterations are explicitly provided. When the target does not belong to the exponential family, the rate of convergence is the same but the asymptotic $L_2$ error increases by a factor $sqrt{ ho^star} > 1$, where $ ho^star - 1$ is the minimum $chi^2$-divergence between the target and an exponential-family proposal.

In Canada, financial advisors and dealers by provincial securities commissions, and those self-regulatory organizations charged with direct regulation over investment dealers and mutual fund dealers, respectively to collect and maintain Know Your Client (KYC) information, such as their age or risk tolerance, for investor accounts. With this information, investors, under their advisor's guidance, make decisions on their investments which are presumed to be beneficial to their investment goals. Our unique dataset is provided by a financial investment dealer with over 50,000 accounts for over 23,000 clients. We use a modified behavioural finance recency, frequency, monetary model for engineering features that quantify investor behaviours, and machine learning clustering algorithms to find groups of investors that behave similarly. We show that the KYC information collected does not explain client behaviours, whereas trade and transaction frequency and volume are most informative. We believe the results shown herein encourage financial regulators and advisors to use more advanced metrics to better understand and predict investor behaviours.

As an important type of reinforcement learning algorithms, actor-critic (AC) and natural actor-critic (NAC) algorithms are often executed in two ways for finding optimal policies. In the first nested-loop design, actor's one update of policy is followed by an entire loop of critic's updates of the value function, and the finite-sample analysis of such AC and NAC algorithms have been recently well established. The second two time-scale design, in which actor and critic update simultaneously but with different learning rates, has much fewer tuning parameters than the nested-loop design and is hence substantially easier to implement. Although two time-scale AC and NAC have been shown to converge in the literature, the finite-sample convergence rate has not been established. In this paper, we provide the first such non-asymptotic convergence rate for two time-scale AC and NAC under Markovian sampling and with actor having general policy class approximation. We show that two time-scale AC requires the overall sample complexity at the order of $mathcal{O}(epsilon^{-2.5}log^3(epsilon^{-1}))$ to attain an $epsilon$-accurate stationary point, and two time-scale NAC requires the overall sample complexity at the order of $mathcal{O}(epsilon^{-4}log^2(epsilon^{-1}))$ to attain an $epsilon$-accurate global optimal point. We develop novel techniques for bounding the bias error of the actor due to dynamically changing Markovian sampling and for analyzing the convergence rate of the linear critic with dynamically changing base functions and transition kernel.

We propose a new semiparametric approach for modelling nonlinear univariate diffusions, where the observed process is a nonparametric transformation of an underlying parametric diffusion (UPD). This modelling strategy yields a general class of semiparametric Markov diffusion models with parametric dynamic copulas and nonparametric marginal distributions. We provide primitive conditions for the identification of the UPD parameters together with the unknown transformations from discrete samples. Likelihood-based estimators of both parametric and nonparametric components are developed and we analyze the asymptotic properties of these. Kernel-based drift and diffusion estimators are also proposed and shown to be normally distributed in large samples. A simulation study investigates the finite sample performance of our estimators in the context of modelling US short-term interest rates. We also present a simple application of the proposed method for modelling the CBOE volatility index data.

The ability of intelligent agents to learn and remember multiple tasks sequentially is crucial to achieving artificial general intelligence. Many continual learning (CL) methods have been proposed to overcome catastrophic forgetting. Catastrophic forgetting notoriously impedes the sequential learning of neural networks as the data of previous tasks are unavailable. In this paper we focus on class incremental learning, a challenging CL scenario, in which classes of each task are disjoint and task identity is unknown during test. For this scenario, generative replay is an effective strategy which generates and replays pseudo data for previous tasks to alleviate catastrophic forgetting. However, it is not trivial to learn a generative model continually for relatively complex data. Based on recently proposed orthogonal weight modification (OWM) algorithm which can keep previously learned input-output mappings invariant approximately when learning new tasks, we propose to directly generate and replay feature. Empirical results on image and text datasets show our method can improve OWM consistently by a significant margin while conventional generative replay always results in a negative effect. Our method also beats a state-of-the-art generative replay method and is competitive with a strong baseline based on real data storage.

In causal settings, such as instrumental variable settings, it is well known that estimators based on ordinary least squares (OLS) can yield biased and non-consistent estimates of the causal parameters. This is partially overcome by two-stage least squares (TSLS) estimators. These are, under weak assumptions, consistent but do not have desirable finite sample properties: in many models, for example, they do not have finite moments. The set of K-class estimators can be seen as a non-linear interpolation between OLS and TSLS and are known to have improved finite sample properties. Recently, in causal discovery, invariance properties such as the moment criterion which TSLS estimators leverage have been exploited for causal structure learning: e.g., in cases, where the causal parameter is not identifiable, some structure of the non-zero components may be identified, and coverage guarantees are available. Subsequently, anchor regression has been proposed to trade-off invariance and predictability. The resulting estimator is shown to have optimal predictive performance under bounded shift interventions. In this paper, we show that the concepts of anchor regression and K-class estimators are closely related. Establishing this connection comes with two benefits: (1) It enables us to prove robustness properties for existing K-class estimators when considering distributional shifts. And, (2), we propose a novel estimator in instrumental variable settings by minimizing the mean squared prediction error subject to the constraint that the estimator lies in an asymptotically valid confidence region of the causal parameter. We call this estimator PULSE (p-uncorrelated least squares estimator) and show that it can be computed efficiently, even though the underlying optimization problem is non-convex. We further prove that it is consistent.

Motion synthesis in a dynamic environment has been a long-standing problem for character animation. Methods using motion capture data tend to scale poorly in complex environments because of their larger capturing and labeling requirement. Physics-based controllers are effective in this regard, albeit less controllable. In this paper, we present CARL, a quadruped agent that can be controlled with high-level directives and react naturally to dynamic environments. Starting with an agent that can imitate individual animation clips, we use Generative Adversarial Networks to adapt high-level controls, such as speed and heading, to action distributions that correspond to the original animations. Further fine-tuning through the deep reinforcement learning enables the agent to recover from unseen external perturbations while producing smooth transitions. It then becomes straightforward to create autonomous agents in dynamic environments by adding navigation modules over the entire process. We evaluate our approach by measuring the agent's ability to follow user control and provide a visual analysis of the generated motion to show its effectiveness.

This paper revisits the problem of computing empirical cumulative distribution functions (ECDF) efficiently on large, multivariate datasets. Computing an ECDF at one evaluation point requires $mathcal{O}(N)$ operations on a dataset composed of $N$ data points. Therefore, a direct evaluation of ECDFs at $N$ evaluation points requires a quadratic $mathcal{O}(N^2)$ operations, which is prohibitive for large-scale problems. Two fast and exact methods are proposed and compared. The first one is based on fast summation in lexicographical order, with a $mathcal{O}(N{log}N)$ complexity and requires the evaluation points to lie on a regular grid. The second one is based on the divide-and-conquer principle, with a $mathcal{O}(Nlog(N)^{(d-1){vee}1})$ complexity and requires the evaluation points to coincide with the input points. The two fast algorithms are described and detailed in the general $d$-dimensional case, and numerical experiments validate their speed and accuracy. Secondly, the paper establishes a direct connection between cumulative distribution functions and kernel density estimation (KDE) for a large class of kernels. This connection paves the way for fast exact algorithms for multivariate kernel density estimation and kernel regression. Numerical tests with the Laplacian kernel validate the speed and accuracy of the proposed algorithms. A broad range of large-scale multivariate density estimation, cumulative distribution estimation, survival function estimation and regression problems can benefit from the proposed numerical methods.

This paper proposes a logistic undirected network formation model which allows for assortative matching on observed individual characteristics and the presence of edge-wise fixed effects. We model the coefficients of observed characteristics to have a latent community structure and the edge-wise fixed effects to be of low rank. We propose a multi-step estimation procedure involving nuclear norm regularization, sample splitting, iterative logistic regression and spectral clustering to detect the latent communities. We show that the latent communities can be exactly recovered when the expected degree of the network is of order log n or higher, where n is the number of nodes in the network. The finite sample performance of the new estimation and inference methods is illustrated through both simulated and real datasets.

In this paper we obtain the limit distribution for partial sums with a random number of terms following a class of mixed Poisson distributions. The resulting weak limit is a mixing between a normal distribution and an exponential family, which we call by normal exponential family (NEF) laws. A new stability concept is introduced and a relationship between {alpha}-stable distributions and NEF laws is established. We propose estimation of the parameters of the NEF models through the method of moments and also by the maximum likelihood method, which is performed via an Expectation-Maximization algorithm. Monte Carlo simulation studies are addressed to check the performance of the proposed estimators and an empirical illustration on financial market is presented.

Sparse estimation via penalized likelihood (PL) is now a popular approach to learn the associations among a large set of variables. This paper describes an R package called lslx that implements PL methods for semi-confirmatory structural equation modeling (SEM). In this semi-confirmatory approach, each model parameter can be specified as free/fixed for theory testing, or penalized for exploration. By incorporating either a L1 or minimax concave penalty, the sparsity pattern of the parameter matrix can be efficiently explored. Package lslx minimizes the PL criterion through a quasi-Newton method. The algorithm conducts line search and checks the first-order condition in each iteration to ensure the optimality of the obtained solution. A numerical comparison between competing packages shows that lslx can reliably find PL estimates with the least time. The current package also supports other advanced functionalities, including a two-stage method with auxiliary variables for missing data handling and a reparameterized multi-group SEM to explore population heterogeneity.

Broadcasting Health and Disease: Bodies, markets and television, 1950s–1980s An ERC BodyCapital international conference to be held at the Wellcome Trust, 19–21 February 2018 In the television age, health and the body have been broadcasted in many ways: in short… Continue reading