In precision medicine, identifying optimal sequences of decision rules, termed dynamic treatment regimes (DTRs), is an important undertaking. One approach investigators may take to infer about optimal DTRs is via Bayesian dynamic Marginal Structural Models (MSMs). These models represent the expected outcome under adherence to a DTR for DTRs in a family indexed by a parameter $ \psi $; the function mapping regimes in the family to the expected outcome under adherence to a DTR is known as the value function. Models that allow for the straightforward identification of an optimal DTR may lead to biased estimates. If such a model is computationally tractable, common wisdom says that a grid-search for the optimal DTR may obviate this difficulty. In a Bayesian context, computational difficulties may be compounded if a posterior mean must be calculated at each grid point. We seek to alleviate these inferential challenges by implementing Gaussian Process ($ \mathcal{GP} $) optimization methods for estimators for the causal effect of adherence to a specified DTR. We examine how to identify optimal DTRs in settings where the value function is multi-modal, which are often not addressed in the DTR literature. We conclude that a $ \mathcal{GP} $ modeling approach that acknowledges noise in the estimated response surface leads to improved results. Additionally, we find that a grid-search may not always yield a robust solution and that it is often less efficient than a $ \mathcal{GP} $ approach. We illustrate the use of the proposed methods by analyzing a clinical dataset with the aim of quantifying the effect of different patterns of HIV therapy.
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We consider a causal inference model in which individuals interact in a social network and they may not comply with the assigned treatments. Estimating causal parameters is challenging in the presence of network interference of unknown form, as each individual may be influenced by both close individuals and distant ones in complex ways. Noncompliance with treatment assignment further complicates this problem, and prior methods dealing with network spillovers but disregarding the noncompliance issue may underestimate the effect of the treatment receipt on the outcome. To estimate meaningful causal parameters, we introduce a new concept of exposure mapping, which summarizes potentially complicated spillover effects into a fixed dimensional statistic of instrumental variables. We investigate identification conditions for the intention-to-treat effect and the average causal effect for compliers, while explicitly considering the possibility of misspecification of exposure mapping. Based on our identification results, we develop nonparametric estimation procedures via inverse probability weighting. Their asymptotic properties, including consistency and asymptotic normality, are investigated using an approximate neighborhood interference framework, which is convenient for dealing with unknown forms of spillovers between individuals. For an empirical illustration, we apply our method to experimental data on the anti-conflict intervention school program.
It is a well-known fact that there is no complete and discrete invariant on the collection of all multiparameter persistence modules. Nonetheless, many invariants have been proposed in the literature to study multiparameter persistence modules, though each invariant will lose some amount of information. One such invariant is the generalized rank invariant. This invariant is known to be complete on the class of interval decomposable persistence modules in general, under mild assumptions on the indexing poset $P$. There is often a trade-off, where the stronger an invariant is, the more expensive it is to compute in practice. The generalized rank invariant on its own is difficult to compute, whereas the standard rank invariant is readily computable through software implementations such as RIVET. We can interpolate between these two to induce new invariants via restricting the domain of the generalized rank invariant, and this family exhibits the aforementioned trade-off. This work studies the tension which exists between computational efficiency and retaining strength when restricting the domain of the generalized rank invariant. We provide a characterization result on where such restrictions are complete invariants in the setting where $P$ is finite, and furthermore show that such restricted generalized rank invariants are stable.
Given a partial differential equation (PDE), goal-oriented error estimation allows us to understand how errors in a diagnostic quantity of interest (QoI), or goal, occur and accumulate in a numerical approximation, for example using the finite element method. By decomposing the error estimates into contributions from individual elements, it is possible to formulate adaptation methods, which modify the mesh with the objective of minimising the resulting QoI error. However, the standard error estimate formulation involves the true adjoint solution, which is unknown in practice. As such, it is common practice to approximate it with an 'enriched' approximation (e.g. in a higher order space or on a refined mesh). Doing so generally results in a significant increase in computational cost, which can be a bottleneck compromising the competitiveness of (goal-oriented) adaptive simulations. The central idea of this paper is to develop a "data-driven" goal-oriented mesh adaptation approach through the selective replacement of the expensive error estimation step with an appropriately configured and trained neural network. In doing so, the error estimator may be obtained without even constructing the enriched spaces. An element-by-element construction is employed here, whereby local values of various parameters related to the mesh geometry and underlying problem physics are taken as inputs, and the corresponding contribution to the error estimator is taken as output. We demonstrate that this approach is able to obtain the same accuracy with a reduced computational cost, for adaptive mesh test cases related to flow around tidal turbines, which interact via their downstream wakes, and where the overall power output of the farm is taken as the QoI. Moreover, we demonstrate that the element-by-element approach implies reasonably low training costs.
Support vector machine (SVM) is a powerful classification method that has achieved great success in many fields. Since its performance can be seriously impaired by redundant covariates, model selection techniques are widely used for SVM with high dimensional covariates. As an alternative to model selection, significant progress has been made in the area of model averaging in the past decades. Yet no frequentist model averaging method was considered for SVM. This work aims to fill the gap and to propose a frequentist model averaging procedure for SVM which selects the optimal weight by cross validation. Even when the number of covariates diverges at an exponential rate of the sample size, we show asymptotic optimality of the proposed method in the sense that the ratio of its hinge loss to the lowest possible loss converges to one. We also derive the convergence rate which provides more insights to model averaging. Compared to model selection methods of SVM which require a tedious but critical task of tuning parameter selection, the model averaging method avoids the task and shows promising performances in the empirical studies.
Causal effect estimation from observational data is a challenging problem, especially with high dimensional data and in the presence of unobserved variables. The available data-driven methods for tackling the problem either provide an estimation of the bounds of a causal effect (i.e. nonunique estimation) or have low efficiency. The major hurdle for achieving high efficiency while trying to obtain unique and unbiased causal effect estimation is how to find a proper adjustment set for confounding control in a fast way, given the huge covariate space and considering unobserved variables. In this paper, we approach the problem as a local search task for finding valid adjustment sets in data. We establish the theorems to support the local search for adjustment sets, and we show that unique and unbiased estimation can be achieved from observational data even when there exist unobserved variables. We then propose a data-driven algorithm that is fast and consistent under mild assumptions. We also make use of a frequent pattern mining method to further speed up the search of minimal adjustment sets for causal effect estimation. Experiments conducted on extensive synthetic and real-world datasets demonstrate that the proposed algorithm outperforms the state-of-the-art criteria/estimators in both accuracy and time-efficiency.
Spatial data can exhibit dependence structures more complicated than can be represented using models that rely on the traditional assumptions of stationarity and isotropy. Several statistical methods have been developed to relax these assumptions. One in particular, the "spatial deformation approach" defines a transformation from the geographic space in which data are observed, to a latent space in which stationarity and isotropy are assumed to hold. Taking inspiration from this class of models, we develop a new model for spatially dependent data observed on graphs. Our method implies an embedding of the graph into Euclidean space wherein the covariance can be modeled using traditional covariance functions such as those from the Mat\'{e}rn family. This is done via a class of graph metrics compatible with such covariance functions. By estimating the edge weights which underlie these metrics, we can recover the "intrinsic distance" between nodes of a graph. We compare our model to existing methods for spatially dependent graph data, primarily conditional autoregressive (CAR) models and their variants and illustrate the advantages our approach has over traditional methods. We fit our model and competitors to bird abundance data for several species in North Carolina. We find that our model fits the data best, and provides insight into the interaction between species-specific spatial distributions and geography.
Cluster-level inference procedures are widely used for brain mapping. These methods compare the size of clusters obtained by thresholding brain maps to an upper bound under the global null hypothesis, computed using Random Field Theory or permutations. However, the guarantees obtained by this type of inference - i.e. at least one voxel is truly activated in the cluster - are not informative with regards to the strength of the signal therein. There is thus a need for methods to assess the amount of signal within clusters; yet such methods have to take into account that clusters are defined based on the data, which creates circularity in the inference scheme. This has motivated the use of post hoc estimates that allow statistically valid estimation of the proportion of activated voxels in clusters. In the context of fMRI data, the All-Resolutions Inference framework introduced in [25] provides post hoc estimates of the proportion of activated voxels. However, this method relies on parametric threshold families, which results in conservative inference. In this paper, we leverage randomization methods to adapt to data characteristics and obtain tighter false discovery control. We obtain Notip, for Non-parametric True Discovery Proportion control: a powerful, non-parametric method that yields statistically valid guarantees on the proportion of activated voxels in data-derived clusters. Numerical experiments demonstrate substantial gains in number of detections compared with state-of-the-art methods on 36 fMRI datasets. The conditions under which the proposed method brings benefits are also discussed.
We establish the minimax risk for parameter estimation in sparse high-dimensional Gaussian mixture models and show that a constrained maximum likelihood estimator (MLE) achieves the minimax optimality. However, the optimization-based constrained MLE is computationally intractable due to non-convexity of the problem. Therefore, we propose a Bayesian approach to estimate high-dimensional Gaussian mixtures whose cluster centers exhibit sparsity using a continuous spike-and-slab prior, and prove that the posterior contraction rate of the proposed Bayesian method is minimax optimal. The mis-clustering rate is obtained as a by-product using tools from matrix perturbation theory. Computationally, posterior inference of the proposed Bayesian method can be implemented via an efficient Gibbs sampler with data augmentation, circumventing the challenging frequentist nonconvex optimization-based algorithms. The proposed Bayesian sparse Gaussian mixture model does not require pre-specifying the number of clusters, which is allowed to grow with the sample size and can be adaptively estimated via posterior inference. The validity and usefulness of the proposed method is demonstrated through simulation studies and the analysis of a real-world single-cell RNA sequencing dataset.
Spatial Gaussian process regression models typically contain finite dimensional covariance parameters that need to be estimated from the data. We study the Bayesian estimation of covariance parameters including the nugget parameter in a general class of stationary covariance functions under fixed-domain asymptotics, which is theoretically challenging due to the increasingly strong dependence among spatial observations. We propose a novel adaptation of the Schwartz's consistency theorem for showing posterior contraction rates of the covariance parameters including the nugget. We derive a new polynomial evidence lower bound, and propose new consistent higher-order quadratic variation estimators that satisfy concentration inequalities with exponentially small tails. Our Bayesian fixed-domain asymptotics theory leads to explicit posterior contraction rates for the microergodic and nugget parameters in the isotropic Matern covariance function under a general stratified sampling design. We verify our theory and the Bayesian predictive performance in simulation studies and an application to sea surface temperature data.
We examine acoustic Doppler current profiler (ADCP) measurements from underwater gliders to determine glider position, glider velocity, and subsurface current. ADCPs, however, do not directly observe the quantities of interest; instead, they measure the relative motion of the vehicle and the water column. We examine the lineage of mathematical innovations that have previously been applied to this problem, discovering an unstated but incorrect assumption of independence. We reframe a recent method to form a joint probability model of current and vehicle navigation, which allows us to correct this assumption and extend the classic Kalman smoothing method. Detailed simulations affirm the efficacy of our approach for computing estimates and their uncertainty. The joint model developed here sets the stage for future work to incorporate constraints, range measurements, and robust statistical modeling.
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