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In this paper we derive a Large Deviation Principle (LDP) for inhomogeneous U/V-statistics of a general order. Using this, we derive a LDP for two types of statistics: random multilinear forms, and number of monochromatic copies of a subgraph. We show that the corresponding rate functions in these cases can be expressed as a variational problem over a suitable space of functions. We use the tools developed to study Gibbs measures with the corresponding Hamiltonians, which include tensor generalizations of both Ising (with non-compact base measure) and Potts models. For these Gibbs measures, we establish scaling limits of log normalizing constants, and weak laws in terms of weak* topology, which are of possible independent interest.

Popular iterative algorithms such as boosting methods and coordinate descent on linear models converge to the maximum $\ell_1$-margin classifier, a.k.a. sparse hard-margin SVM, in high dimensional regimes where the data is linearly separable. Previous works consistently show that many estimators relying on the $\ell_1$-norm achieve improved statistical rates for hard sparse ground truths. We show that surprisingly, this adaptivity does not apply to the maximum $\ell_1$-margin classifier for a standard discriminative setting. In particular, for the noiseless setting, we prove tight upper and lower bounds for the prediction error that match existing rates of order $\frac{\|\wgt\|_1^{2/3}}{n^{1/3}}$ for general ground truths. To complete the picture, we show that when interpolating noisy observations, the error vanishes at a rate of order $\frac{1}{\sqrt{\log(d/n)}}$. We are therefore first to show benign overfitting for the maximum $\ell_1$-margin classifier.

Prescriptive process monitoring methods seek to improve the performance of a process by selectively triggering interventions at runtime (e.g., offering a discount to a customer) to increase the probability of a desired case outcome (e.g., a customer making a purchase). The backbone of a prescriptive process monitoring method is an intervention policy, which determines for which cases and when an intervention should be executed. Existing methods in this field rely on predictive models to define intervention policies; specifically, they consider policies that trigger an intervention when the estimated probability of a negative outcome exceeds a threshold. However, the probabilities computed by a predictive model may come with a high level of uncertainty (low confidence), leading to unnecessary interventions and, thus, wasted effort. This waste is particularly problematic when the resources available to execute interventions are limited. To tackle this shortcoming, this paper proposes an approach to extend existing prescriptive process monitoring methods with so-called conformal predictions, i.e., predictions with confidence guarantees. An empirical evaluation using real-life public datasets shows that conformal predictions enhance the net gain of prescriptive process monitoring methods under limited resources.

Via operator theoretic methods, we formalize the concentration phenomenon for a given observable `$r$' of a discrete time Markov chain with `$\mu_{\pi}$' as invariant ergodic measure, possibly having support on an unbounded state space. The main contribution of this paper is circumventing tedious probabilistic methods with a study of a composition of the Markov transition operator $P$ followed by a multiplication operator defined by $e^{r}$. It turns out that even if the observable/ reward function is unbounded, but for some for some $q>2$, $\|e^{r}\|_{q \rightarrow 2} \propto \exp\big(\mu_{\pi}(r) +\frac{2q}{q-2}\big) $ and $P$ is hyperbounded with norm control $\|P\|_{2 \rightarrow q }< e^{\frac{1}{2}[\frac{1}{2}-\frac{1}{q}]}$, sharp non-asymptotic concentration bounds follow. \emph{Transport-entropy} inequality ensures the aforementioned upper bound on multiplication operator for all $q>2$. The role of \emph{reversibility} in concentration phenomenon is demystified. These results are particularly useful for the reinforcement learning and controls communities as they allow for concentration inequalities w.r.t standard unbounded obersvables/reward functions where exact knowledge of the system is not available, let alone the reversibility of stationary measure.

Missing data can lead to inefficiencies and biases in analyses, in particular when data are missing not at random (MNAR). It is thus vital to understand and correctly identify the missing data mechanism. Recovering missing values through a follow up sample allows researchers to conduct hypothesis tests for MNAR, which are not possible when using only the original incomplete data. Investigating how properties of these tests are affected by the follow up sample design is little explored in the literature. Our results provide comprehensive insight into the properties of one such test, based on the commonly used selection model framework. We determine conditions for recovery samples that allow the test to be applied appropriately and effectively, i.e. with known Type I error rates and optimized with respect to power. We thus provide an integrated framework for testing for the presence of MNAR and designing follow up samples in an efficient cost-effective way. The performance of our methodology is evaluated through simulation studies as well as on a real data sample.

In social, medical, and behavioral research we often encounter datasets with a multilevel structure and multiple correlated dependent variables. These data are frequently collected from a study population that distinguishes several subpopulations with different (i.e. heterogeneous) effects of an intervention. Despite the frequent occurrence of such data, methods to analyze them are less common and researchers often resort to either ignoring the multilevel and/or heterogeneous structure, analyzing only a single dependent variable, or a combination of these. These analysis strategies are suboptimal: Ignoring multilevel structures inflates Type I error rates, while neglecting the multivariate or heterogeneous structure masks detailed insights. To analyze such data comprehensively, the current paper presents a novel Bayesian multilevel multivariate logistic regression model. The clustered structure of multilevel data is taken into account, such that posterior inferences can be made with accurate error rates. Further, the model shares information between different subpopulations in the estimation of average and conditional average multivariate treatment effects. To facilitate interpretation, multivariate logistic regression parameters are transformed to posterior success probabilities and differences between them. A numerical evaluation compared our framework to less comprehensive alternatives and highlighted the need to model the multilevel structure: Treatment comparisons based on the multilevel model had targeted Type I error rates, while single-level alternatives resulted in inflated Type I errors. A re-analysis of the Third International Stroke Trial data illustrated how incorporating a multilevel structure, assessing treatment heterogeneity, and combining dependent variables contributed to an in-depth understanding of treatment effects.

This paper considers the problem of kernel regression and classification with possibly unobservable response variables in the data, where the mechanism that causes the absence of information is unknown and can depend on both predictors and the response variables. Our proposed approach involves two steps: In the first step, we construct a family of models (possibly infinite dimensional) indexed by the unknown parameter of the missing probability mechanism. In the second step, a search is carried out to find the empirically optimal member of an appropriate cover (or subclass) of the underlying family in the sense of minimizing the mean squared prediction error. The main focus of the paper is to look into the theoretical properties of these estimators. The issue of identifiability is also addressed. Our methods use a data-splitting approach which is quite easy to implement. We also derive exponential bounds on the performance of the resulting estimators in terms of their deviations from the true regression curve in general Lp norms, where we also allow the size of the cover or subclass to diverge as the sample size n increases. These bounds immediately yield various strong convergence results for the proposed estimators. As an application of our findings, we consider the problem of statistical classification based on the proposed regression estimators and also look into their rates of convergence under different settings. Although this work is mainly stated for kernel-type estimators, they can also be extended to other popular local-averaging methods such as nearest-neighbor estimators, and histogram estimators.

To explore the limits of a stochastic gradient method, it may be useful to consider an example consisting of an infinite number of quadratic functions. In this context, it is appropriate to determine the expected value and the covariance matrix of the stochastic noise, i.e. the difference of the true gradient and the approximated gradient generated from a finite sample. When specifying the covariance matrix, the expected value of a quadratic form QBQ is needed, where Q is a Wishart distributed random matrix and B is an arbitrary fixed symmetric matrix. After deriving an expression for E(QBQ) and considering some special cases, a numerical example is used to show how these results can support the comparison of two stochastic methods.

We consider the following data perturbation model, where the covariates incur multiplicative errors. For two $n \times m$ random matrices $U, X$, we denote by $U \circ X$ the Hadamard or Schur product, which is defined as $(U \circ X)_{ij} = (U_{ij}) \cdot (X_{ij})$. In this paper, we study the subgaussian matrix variate model, where we observe the matrix variate data $X$ through a random mask $U$: $$ {\mathcal X} = U \circ X \; \; \; \text{ where} \; \; \;X = B^{1/2} {\mathbb{Z}} A^{1/2}, $$ where ${\mathbb{Z}}$ is a random matrix with independent subgaussian entries, and $U$ is a mask matrix with either zero or positive entries, where ${\mathbb E} U_{ij} \in [0, 1]$ and all entries are mutually independent. Subsampling in rows, or columns, or random sampling of entries of $X$ are special cases of this model. Under the assumption of independence between $U$ and $X$, we introduce componentwise unbiased estimators for estimating covariance $A$ and $B$, and prove the concentration of measure bounds in the sense of guaranteeing the restricted eigenvalue($\textsf{RE}$) conditions to hold on the unbiased estimator for $B$, when columns of data matrix $X$ are sampled with different rates. We further develop multiple regression methods for estimating the inverse of $B$ and show statistical rate of convergence. Our results provide insight for sparse recovery for relationships among entities (samples, locations, items) when features (variables, time points, user ratings) are present in the observed data matrix ${\mathcal X}$ with heterogeneous rates. Our proof techniques can certainly be extended to other scenarios. We provide simulation evidence illuminating the theoretical predictions.

For policymakers wishing to make evidence-based decisions, one of the challenges is how to combine the relevant information and evidence in a coherent and defensible manner in order to formulate and evaluate candidate policies. Policymakers often need to rely on experts with disparate fields of expertise when making policy choices in complex, multi-faceted, dynamic environments such as those dealing with ecosystem services. The pressures affecting the survival and pollination capabilities of honey bees (Apis mellifera), wild bees and other pollinators is well-documented, but incomplete. In order to estimate the potential effectiveness of various candidate policies to support pollination services, there is an urgent need to quantify the effect of various combinations of variables on the pollination ecosystem service, utilising available information, models and expert judgement. In this paper, we present a new application of the integrating decision support system methodology for combining inputs from multiple panels of experts to evaluate policies to support an abundant pollinator population.