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Register a new userConditional Mutual Information Estimation for Mixed Discrete and Continuous Variables with Nearest Neighbors
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Added by
Octavio Mesner
Publication date
2019
fields
Mathematical Statistics
and research's language is
English
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Fields like public health, public policy, and social science often want to quantify the degree of dependence between variables whose relationships take on unknown functional forms. Typically, in fact, researchers in these fields are attempting to evaluate causal theories, and so want to quantify dependence after conditioning on other variables that might explain, mediate or confound causal relations. One reason conditional mutual information is not more widely used for these tasks is the lack of estimators which can handle combinations of continuous and discrete random variables, common in applications. This paper develops a new method for estimating mutual and conditional mutual information for data samples containing a mix of discrete and continuous variables. We prove that this estimator is consistent and show, via simulation, that it is more accurate than similar estimators.
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Nonparametric estimation of mutual information is used in a wide range of scientific problems to quantify dependence between variables. The k-nearest neighbor (knn) methods are consistent, and therefore expected to work well for large sample size. These methods use geometrically regular local volume elements. This practice allows maximum localization of the volume elements, but can also induce a bias due to a poor description of the local geometry of the underlying probability measure. We introduce a new class of knn estimators that we call geometric knn estimators (g-knn), which use more complex local volume elements to better model the local geometry of the probability measures. As an example of this class of estimators, we develop a g-knn estimator of entropy and mutual information based on elliptical volume elements, capturing the local stretching and compression common to a wide range of dynamical systems attractors. A series of numerical examples in which the thickness of the underlying distribution and the sample sizes are varied suggest that local geometry is a source of problems for knn methods such as the Kraskov-St{o}gbauer-Grassberger (KSG) estimator when local geometric effects cannot be removed by global preprocessing of the data. The g-knn method performs well despite the manipulation of the local geometry. In addition, the examples suggest that the g-knn estimators can be of particular relevance to applications in which the system is large, but data size is limited.
We develop a Nonparametric Empirical Bayes (NEB) framework for compound estimation in the discrete linear exponential family, which includes a wide class of discrete distributions frequently arising from modern big data applications. We propose to directly estimate the Bayes shrinkage factor in the generalized Robbins formula via solving a scalable convex program, which is carefully developed based on a RKHS representation of the Steins discrepancy measure. The new NEB estimation framework is flexible for incorporating various structural constraints into the data driven rule, and provides a unified approach to compound estimation with both regular and scaled squared error losses. We develop theory to show that the class of NEB estimators enjoys strong asymptotic properties. Comprehensive simulation studies as well as analyses of real data examples are carried out to demonstrate the superiority of the NEB estimator over competing methods.
This paper studies the generalization of the targeted minimum loss-based estimation (TMLE) framework to estimation of effects of time-varying interventions in settings where both interventions, covariates, and outcome can happen at subject-specific time-points on an arbitrarily fine time-scale. TMLE is a general template for constructing asymptotically linear substitution estimators for smooth low-dimensional parameters in infinite-dimensional models. Existing longitudinal TMLE methods are developed for data where observations are made on a discrete time-grid. We consider a continuous-time counting process model where intensity measures track the monitoring of subjects, and focus on a low-dimensional target parameter defined as the intervention-specific mean outcome at the end of follow-up. To construct our TMLE algorithm for the given statistical estimation problem we derive an expression for the efficient influence curve and represent the target parameter as a functional of intensities and conditional expectations. The high-dimensional nuisance parameters of our model are estimated and updated in an iterative manner according to separate targeting steps for the involved intensities and conditional expectations. The resulting estimator solves the efficient influence curve equation. We state a general efficiency theorem and describe a highly adaptive lasso estimator for nuisance parameters that allows us to establish asymptotic linearity and efficiency of our estimator under minimal conditions on the underlying statistical model.
We study a functional linear regression model that deals with functional responses and allows for both functional covariates and high-dimensional vector covariates. The proposed model is flexible and nests several functional regression models in the literature as special cases. Based on the theory of reproducing kernel Hilbert spaces (RKHS), we propose a penalized least squares estimator that can accommodate functional variables observed on discrete sample points. Besides a conventional smoothness penalty, a group Lasso-type penalty is further imposed to induce sparsity in the high-dimensional vector predictors. We derive finite sample theoretical guarantees and show that the excess prediction risk of our estimator is minimax optimal. Furthermore, our analysis reveals an interesting phase transition phenomenon that the optimal excess risk is determined jointly by the smoothness and the sparsity of the functional regression coefficients. A novel efficient optimization algorithm based on iterative coordinate descent is devised to handle the smoothness and group penalties simultaneously. Simulation studies and real data applications illustrate the promising performance of the proposed approach compared to the state-of-the-art methods in the literature.
Selective recruitment designs preferentially recruit individuals that are estimated to be statistically informative onto a clinical trial. Individuals that are expected to contribute less information have a lower probability of recruitment. Furthermore, in an information-adaptive design recruits are allocated to treatment arms in a manner that maximises information gain. The informativeness of an individual depends on their covariate (or biomarker) values and how information is defined is a critical element of information-adaptive designs. In this paper we define and evaluate four different methods for quantifying statistical information. Using both experimental data and numerical simulations we show that selective recruitment designs can offer a substantial increase in statistical power compared to randomised designs. In trials without selective recruitment we find that allocating individuals to treatment arms according to information-adaptive protocols also leads to an increase in statistical power. Consequently, selective recruitment designs can potentially achieve successful trials using fewer recruits thereby offering economic and ethical advantages.
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