This paper studies a service system in which arriving customers are provided with information about the delay they will experience. Based on this information they decide to wait for service or to leave the system. The main objective is to estimate the customers' patience-level distribution and the corresponding potential arrival rate, using knowledge of the actual queue-length process only. The main complication, and distinguishing feature of our setup, lies in the fact that customers who decide not to join are not observed, but, remarkably, we manage to devise a procedure to estimate the load they would generate. We express our system in terms of a multi-server queue with a Poisson stream of customers, which allows us to evaluate the corresponding likelihood function. Estimating the unknown parameters relying on a maximum likelihood procedure, we prove strong consistency and derive the asymptotic distribution of the estimation error. Several applications and extensions of the method are discussed. The performance of our approach is further assessed through a series of numerical experiments. By fitting parameters of hyperexponential and generalized-hyperexponential distributions our method provides a robust estimation framework for any continuous patience-level distribution.
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The construction of effective Recommender Systems (RS) is a complex process, mainly due to the nature of RSs which involves large scale software-systems and human interactions. Iterative development processes require deep understanding of a current baseline as well as the ability to estimate the impact of changes in multiple variables of interest. Simulations are well suited to address both challenges and potentially leading to a high velocity construction process, a fundamental requirement in commercial contexts. Recently, there has been significant interest in RS Simulation Platforms, which allow RS developers to easily craft simulated environments where their systems can be analysed. In this work we discuss how simulations help to increase velocity, we look at the literature around RS Simulation Platforms, analyse strengths and gaps and distill a set of guiding principles for the design of RS Simulation Platforms that we believe will maximize the velocity of iterative RS construction processes.
Dynamic treatment regimes (DTRs) consist of a sequence of decision rules, one per stage of intervention, that finds effective treatments for individual patients according to patient information history. DTRs can be estimated from models which include the interaction between treatment and a small number of covariates which are often chosen a priori. However, with increasingly large and complex data being collected, it is difficult to know which prognostic factors might be relevant in the treatment rule. Therefore, a more data-driven approach of selecting these covariates might improve the estimated decision rules and simplify models to make them easier to interpret. We propose a variable selection method for DTR estimation using penalized dynamic weighted least squares. Our method has the strong heredity property, that is, an interaction term can be included in the model only if the corresponding main terms have also been selected. Through simulations, we show our method has both the double robustness property and the oracle property, and the newly proposed methods compare favorably with other variable selection approaches.
In many applications, it is of interest to assess the relative contribution of features (or subsets of features) toward the goal of predicting a response -- in other words, to gauge the variable importance of features. Most recent work on variable importance assessment has focused on describing the importance of features within the confines of a given prediction algorithm. However, such assessment does not necessarily characterize the prediction potential of features, and may provide a misleading reflection of the intrinsic value of these features. To address this limitation, we propose a general framework for nonparametric inference on interpretable algorithm-agnostic variable importance. We define variable importance as a population-level contrast between the oracle predictiveness of all available features versus all features except those under consideration. We propose a nonparametric efficient estimation procedure that allows the construction of valid confidence intervals, even when machine learning techniques are used. We also outline a valid strategy for testing the null importance hypothesis. Through simulations, we show that our proposal has good operating characteristics, and we illustrate its use with data from a study of an antibody against HIV-1 infection.
Given a function $u\in L^2=L^2(D,\mu)$, where $D\subset \mathbb R^d$ and $\mu$ is a measure on $D$, and a linear subspace $V_n\subset L^2$ of dimension $n$, we show that near-best approximation of $u$ in $V_n$ can be computed from a near-optimal budget of $Cn$ pointwise evaluations of $u$, with $C>1$ a universal constant. The sampling points are drawn according to some random distribution, the approximation is computed by a weighted least-squares method, and the error is assessed in expected $L^2$ norm. This result improves on the results in [6,8] which require a sampling budget that is sub-optimal by a logarithmic factor, thanks to a sparsification strategy introduced in [17,18]. As a consequence, we obtain for any compact class $\mathcal K\subset L^2$ that the sampling number $\rho_{Cn}^{\rm rand}(\mathcal K)_{L^2}$ in the randomized setting is dominated by the Kolmogorov $n$-width $d_n(\mathcal K)_{L^2}$. While our result shows the existence of a randomized sampling with such near-optimal properties, we discuss remaining issues concerning its generation by a computationally efficient algorithm.
Selection bias is prevalent in the data for training and evaluating recommendation systems with explicit feedback. For example, users tend to rate items they like. However, when rating an item concerning a specific user, most of the recommendation algorithms tend to rely too much on his/her rating (feedback) history. This introduces implicit bias on the recommendation system, which is referred to as user feedback-loop bias in this paper. We propose a systematic and dynamic way to correct such bias and to obtain more diverse and objective recommendations by utilizing temporal rating information. Specifically, our method includes a deep-learning component to learn each user's dynamic rating history embedding for the estimation of the probability distribution of the items that the user rates sequentially. These estimated dynamic exposure probabilities are then used as propensity scores to train an inverse-propensity-scoring (IPS) rating predictor. We empirically validated the existence of such user feedback-loop bias in real world recommendation systems and compared the performance of our method with the baseline models that are either without de-biasing or with propensity scores estimated by other methods. The results show the superiority of our approach.
Consider the task of estimating a 3-order $n \times n \times n$ tensor from noisy observations of randomly chosen entries in the sparse regime. We introduce a similarity based collaborative filtering algorithm for sparse tensor estimation and argue that it achieves sample complexity that nearly matches the conjectured computationally efficient lower bound on the sample complexity for the setting of low-rank tensors. Our algorithm uses the matrix obtained from the flattened tensor to compute similarity, and estimates the tensor entries using a nearest neighbor estimator. We prove that the algorithm recovers a low rank tensor with maximum entry-wise error (MEE) and mean-squared-error (MSE) decaying to $0$ as long as each entry is observed independently with probability $p = \Omega(n^{-3/2 + \kappa})$ for any arbitrarily small $\kappa > 0$. % as long as tensor has finite rank $r = \Theta(1)$. More generally, we establish robustness of the estimator, showing that when arbitrary noise bounded by $\epsilon \geq 0$ is added to each observation, the estimation error with respect to MEE and MSE degrades by ${\sf poly}(\epsilon)$. Consequently, even if the tensor may not have finite rank but can be approximated within $\epsilon \geq 0$ by a finite rank tensor, then the estimation error converges to ${\sf poly}(\epsilon)$. Our analysis sheds insight into the conjectured sample complexity lower bound, showing that it matches the connectivity threshold of the graph used by our algorithm for estimating similarity between coordinates.
Physics informed neural networks (PINNs) have recently been widely used for robust and accurate approximation of PDEs. We provide rigorous upper bounds on the generalization error of PINNs approximating solutions of the forward problem for PDEs. An abstract formalism is introduced and stability properties of the underlying PDE are leveraged to derive an estimate for the generalization error in terms of the training error and number of training samples. This abstract framework is illustrated with several examples of nonlinear PDEs. Numerical experiments, validating the proposed theory, are also presented.
This paper gives a new approach for the maximum likelihood estimation of the joint of the location and scale of the Cauchy distribution. We regard the joint as a single complex parameter and derive a new form of the likelihood equation of a complex variable. Based on the equation, we provide a new iterative scheme approximating the maximum likelihood estimate. We also handle the equation in an algebraic manner and derive a polynomial containing the maximum likelihood estimate as a root. This algebraic approach provides another scheme approximating the maximum likelihood estimate by root-finding algorithms for polynomials, and furthermore, gives non-existence of closed-form formulae for the case that the sample size is five. We finally provide some numerical examples to show our method is effective.
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.
We propose a new method of estimation in topic models, that is not a variation on the existing simplex finding algorithms, and that estimates the number of topics K from the observed data. We derive new finite sample minimax lower bounds for the estimation of A, as well as new upper bounds for our proposed estimator. We describe the scenarios where our estimator is minimax adaptive. Our finite sample analysis is valid for any number of documents (n), individual document length (N_i), dictionary size (p) and number of topics (K), and both p and K are allowed to increase with n, a situation not handled well by previous analyses. We complement our theoretical results with a detailed simulation study. We illustrate that the new algorithm is faster and more accurate than the current ones, although we start out with a computational and theoretical disadvantage of not knowing the correct number of topics K, while we provide the competing methods with the correct value in our simulations.
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