Continuous determinantal point processes (DPPs) are a class of repulsive point processes on $\mathbb{R}^d$ with many statistical applications. Although an explicit expression of their density is known, it is too complicated to be used directly for maximum likelihood estimation. In the stationary case, an approximation using Fourier series has been suggested, but it is limited to rectangular observation windows and no theoretical results support it. In this contribution, we investigate a different way to approximate the likelihood by looking at its asymptotic behaviour when the observation window grows towards $\mathbb{R}^d$. This new approximation is not limited to rectangular windows, is faster to compute than the previous one, does not require any tuning parameter, and some theoretical justifications are provided. It moreover provides an explicit formula for estimating the asymptotic variance of the associated estimator. The performances are assessed in a simulation study on standard parametric models on $\mathbb{R}^d$ and compare favourably to common alternative estimation methods for continuous DPPs.
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We introduce a new distortion measure for point processes called functional-covering distortion. It is inspired by intensity theory and is related to both the covering of point processes and logarithmic loss distortion. We obtain the distortion-rate function with feedforward under this distortion measure for a large class of point processes. For Poisson processes, the rate-distortion function is obtained under a general condition called constrained functional-covering distortion, of which both covering and functional-covering are special cases. Also for Poisson processes, we characterize the rate-distortion region for a two-encoder CEO problem and show that feedforward does not enlarge this region.
A determinantal point process (DPP) on a collection of $M$ items is a model, parameterized by a symmetric kernel matrix, that assigns a probability to every subset of those items. Recent work shows that removing the kernel symmetry constraint, yielding nonsymmetric DPPs (NDPPs), can lead to significant predictive performance gains for machine learning applications. However, existing work leaves open the question of scalable NDPP sampling. There is only one known DPP sampling algorithm, based on Cholesky decomposition, that can directly apply to NDPPs as well. Unfortunately, its runtime is cubic in $M$, and thus does not scale to large item collections. In this work, we first note that this algorithm can be transformed into a linear-time one for kernels with low-rank structure. Furthermore, we develop a scalable sublinear-time rejection sampling algorithm by constructing a novel proposal distribution. Additionally, we show that imposing certain structural constraints on the NDPP kernel enables us to bound the rejection rate in a way that depends only on the kernel rank. In our experiments we compare the speed of all of these samplers for a variety of real-world tasks.
We employ kernel-based approaches that use samples from a probability distribution to approximate a Kolmogorov operator on a manifold. The self-tuning variable-bandwidth kernel method [Berry & Harlim, Appl. Comput. Harmon. Anal., 40(1):68--96, 2016] computes a large, sparse matrix that approximates the differential operator. Here, we use the eigendecomposition of the discretization to (i) invert the operator, solving a differential equation, and (ii) represent gradient vector fields on the manifold. These methods only require samples from the underlying distribution and, therefore, can be applied in high dimensions or on geometrically complex manifolds when spatial discretizations are not available. We also employ an efficient $k$-$d$ tree algorithm to compute the sparse kernel matrix, which is a computational bottleneck.
We provide a decision theoretic analysis of bandit experiments. The setting corresponds to a dynamic programming problem, but solving this directly is typically infeasible. Working within the framework of diffusion asymptotics, we define suitable notions of asymptotic Bayes and minimax risk for bandit experiments. For normally distributed rewards, the minimal Bayes risk can be characterized as the solution to a nonlinear second-order partial differential equation (PDE). Using a limit of experiments approach, we show that this PDE characterization also holds asymptotically under both parametric and non-parametric distribution of the rewards. The approach further describes the state variables it is asymptotically sufficient to restrict attention to, and therefore suggests a practical strategy for dimension reduction. The upshot is that we can approximate the dynamic programming problem defining the bandit experiment with a PDE which can be efficiently solved using sparse matrix routines. We derive the optimal Bayes and minimax policies from the numerical solutions to these equations. The proposed policies substantially dominate existing methods such as Thompson sampling. The framework also allows for substantial generalizations to the bandit problem such as time discounting and pure exploration motives.
We consider M-estimation problems, where the target value is determined using a minimizer of an expected functional of a Levy process. With discrete observations from the Levy process, we can produce a "quasi-path" by shuffling increments of the Levy process, we call it a quasi-process. Under a suitable sampling scheme, a quasi-process can converge weakly to the true process according to the properties of the stationary and independent increments. Using this resampling technique, we can estimate objective functionals similar to those estimated using the Monte Carlo simulations, and it is available as a contrast function. The M-estimator based on these quasi-processes can be consistent and asymptotically normal.
The instrumental variable method is widely used in the health and social sciences for identification and estimation of causal effects in the presence of potentially unmeasured confounding. In order to improve efficiency, multiple instruments are routinely used, leading to concerns about bias due to possible violation of the instrumental variable assumptions. To address this concern, we introduce a new class of g-estimators that are guaranteed to remain consistent and asymptotically normal for the causal effect of interest provided that a set of at least $\gamma$ out of $K$ candidate instruments are valid, for $\gamma\leq K$ set by the analyst ex ante, without necessarily knowing the identities of the valid and invalid instruments. We provide formal semiparametric efficiency theory supporting our results. Both simulation studies and applications to the UK Biobank data demonstrate the superior empirical performance of our estimators compared to competing methods.
Let $X^{(n)}$ be an observation sampled from a distribution $P_{\theta}^{(n)}$ with an unknown parameter $\theta,$ $\theta$ being a vector in a Banach space $E$ (most often, a high-dimensional space of dimension $d$). We study the problem of estimation of $f(\theta)$ for a functional $f:E\mapsto {\mathbb R}$ of some smoothness $s>0$ based on an observation $X^{(n)}\sim P_{\theta}^{(n)}.$ Assuming that there exists an estimator $\hat \theta_n=\hat \theta_n(X^{(n)})$ of parameter $\theta$ such that $\sqrt{n}(\hat \theta_n-\theta)$ is sufficiently close in distribution to a mean zero Gaussian random vector in $E,$ we construct a functional $g:E\mapsto {\mathbb R}$ such that $g(\hat \theta_n)$ is an asymptotically normal estimator of $f(\theta)$ with $\sqrt{n}$ rate provided that $s>\frac{1}{1-\alpha}$ and $d\leq n^{\alpha}$ for some $\alpha\in (0,1).$ We also derive general upper bounds on Orlicz norm error rates for estimator $g(\hat \theta)$ depending on smoothness $s,$ dimension $d,$ sample size $n$ and the accuracy of normal approximation of $\sqrt{n}(\hat \theta_n-\theta).$ In particular, this approach yields asymptotically efficient estimators in some high-dimensional exponential models.
In this paper we study the finite sample and asymptotic properties of various weighting estimators of the local average treatment effect (LATE), several of which are based on Abadie (2003)'s kappa theorem. Our framework presumes a binary endogenous explanatory variable ("treatment") and a binary instrumental variable, which may only be valid after conditioning on additional covariates. We argue that one of the Abadie estimators, which we show is weight normalized, is likely to dominate the others in many contexts. A notable exception is in settings with one-sided noncompliance, where certain unnormalized estimators have the advantage of being based on a denominator that is bounded away from zero. We use a simulation study and three empirical applications to illustrate our findings. In applications to causal effects of college education using the college proximity instrument (Card, 1995) and causal effects of childbearing using the sibling sex composition instrument (Angrist and Evans, 1998), the unnormalized estimates are clearly unreasonable, with "incorrect" signs, magnitudes, or both. Overall, our results suggest that (i) the relative performance of different kappa weighting estimators varies with features of the data-generating process; and that (ii) the normalized version of Tan (2006)'s estimator may be an attractive alternative in many contexts. Applied researchers with access to a binary instrumental variable should also consider covariate balancing or doubly robust estimators of the LATE.
One of the most important problems in system identification and statistics is how to estimate the unknown parameters of a given model. Optimization methods and specialized procedures, such as Empirical Minimization (EM) can be used in case the likelihood function can be computed. For situations where one can only simulate from a parametric model, but the likelihood is difficult or impossible to evaluate, a technique known as the Two-Stage (TS) Approach can be applied to obtain reliable parametric estimates. Unfortunately, there is currently a lack of theoretical justification for TS. In this paper, we propose a statistical decision-theoretical derivation of TS, which leads to Bayesian and Minimax estimators. We also show how to apply the TS approach on models for independent and identically distributed samples, by computing quantiles of the data as a first step, and using a linear function as the second stage. The proposed method is illustrated via numerical simulations.
We recall some of the history of the information-theoretic approach to deriving core results in probability theory and indicate parts of the recent resurgence of interest in this area with current progress along several interesting directions. Then we give a new information-theoretic proof of a finite version of de Finetti's classical representation theorem for finite-valued random variables. We derive an upper bound on the relative entropy between the distribution of the first $k$ in a sequence of $n$ exchangeable random variables, and an appropriate mixture over product distributions. The mixing measure is characterised as the law of the empirical measure of the original sequence, and de Finetti's result is recovered as a corollary. The proof is nicely motivated by the Gibbs conditioning principle in connection with statistical mechanics, and it follows along an appealing sequence of steps. The technical estimates required for these steps are obtained via the use of a collection of combinatorial tools known within information theory as `the method of types.'
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