Søren Wengel Mogensen, Niels Richard Hansen.

Source: The Annals of Statistics, Volume 48, Number 1, 539--559.

Abstract:
Symmetric independence relations are often studied using graphical representations. Ancestral graphs or acyclic directed mixed graphs with $m$-separation provide classes of symmetric graphical independence models that are closed under marginalization. Asymmetric independence relations appear naturally for multivariate stochastic processes, for instance, in terms of local independence. However, no class of graphs representing such asymmetric independence relations, which is also closed under marginalization, has been developed. We develop the theory of directed mixed graphs with $mu $-separation and show that this provides a graphical independence model class which is closed under marginalization and which generalizes previously considered graphical representations of local independence. Several graphs may encode the same set of independence relations and this means that in many cases only an equivalence class of graphs can be identified from observational data. For statistical applications, it is therefore pivotal to characterize graphs that induce the same independence relations. Our main result is that for directed mixed graphs with $mu $-separation each equivalence class contains a maximal element which can be constructed from the independence relations alone. Moreover, we introduce the directed mixed equivalence graph as the maximal graph with dashed and solid edges. This graph encodes all information about the edges that is identifiable from the independence relations, and furthermore it can be computed efficiently from the maximal graph.

Edgar Dobriban, William Leeb, Amit Singer.

Source: The Annals of Statistics, Volume 48, Number 1, 491--513.

Abstract:
We consider the linearly transformed spiked model , where the observations $Y_{i}$ are noisy linear transforms of unobserved signals of interest $X_{i}$: egin{equation*}Y_{i}=A_{i}X_{i}+varepsilon_{i},end{equation*} for $i=1,ldots ,n$. The transform matrices $A_{i}$ are also observed. We model the unobserved signals (or regression coefficients) $X_{i}$ as vectors lying on an unknown low-dimensional space. Given only $Y_{i}$ and $A_{i}$ how should we predict or recover their values? The naive approach of performing regression for each observation separately is inaccurate due to the large noise level. Instead, we develop optimal methods for predicting $X_{i}$ by “borrowing strength” across the different samples. Our linear empirical Bayes methods scale to large datasets and rely on weak moment assumptions. We show that this model has wide-ranging applications in signal processing, deconvolution, cryo-electron microscopy, and missing data with noise. For missing data, we show in simulations that our methods are more robust to noise and to unequal sampling than well-known matrix completion methods.

Vladimir Koltchinskii, Matthias Löffler, Richard Nickl.

Source: The Annals of Statistics, Volume 48, Number 1, 464--490.

Abstract:
We study principal component analysis (PCA) for mean zero i.i.d. Gaussian observations $X_{1},dots,X_{n}$ in a separable Hilbert space $mathbb{H}$ with unknown covariance operator $Sigma $. The complexity of the problem is characterized by its effective rank $mathbf{r}(Sigma):=frac{operatorname{tr}(Sigma)}{|Sigma |}$, where $mathrm{tr}(Sigma)$ denotes the trace of $Sigma $ and $|Sigma|$ denotes its operator norm. We develop a method of bias reduction in the problem of estimation of linear functionals of eigenvectors of $Sigma $. Under the assumption that $mathbf{r}(Sigma)=o(n)$, we establish the asymptotic normality and asymptotic properties of the risk of the resulting estimators and prove matching minimax lower bounds, showing their semiparametric optimality.

François Bachoc, David Preinerstorfer, Lukas Steinberger.

Source: The Annals of Statistics, Volume 48, Number 1, 440--463.

Abstract:
We suggest general methods to construct asymptotically uniformly valid confidence intervals post-model-selection. The constructions are based on principles recently proposed by Berk et al. ( Ann. Statist. 41 (2013) 802–837). In particular, the candidate models used can be misspecified, the target of inference is model-specific, and coverage is guaranteed for any data-driven model selection procedure. After developing a general theory, we apply our methods to practically important situations where the candidate set of models, from which a working model is selected, consists of fixed design homoskedastic or heteroskedastic linear models, or of binary regression models with general link functions. In an extensive simulation study, we find that the proposed confidence intervals perform remarkably well, even when compared to existing methods that are tailored only for specific model selection procedures.

Changliang Zou, Guanghui Wang, Runze Li.

Source: The Annals of Statistics, Volume 48, Number 1, 413--439.

Abstract:
In multiple change-point analysis, one of the major challenges is to estimate the number of change-points. Most existing approaches attempt to minimize a Schwarz information criterion which balances a term quantifying model fit with a penalization term accounting for model complexity that increases with the number of change-points and limits overfitting. However, different penalization terms are required to adapt to different contexts of multiple change-point problems and the optimal penalization magnitude usually varies from the model and error distribution. We propose a data-driven selection criterion that is applicable to most kinds of popular change-point detection methods, including binary segmentation and optimal partitioning algorithms. The key idea is to select the number of change-points that minimizes the squared prediction error, which measures the fit of a specified model for a new sample. We develop a cross-validation estimation scheme based on an order-preserved sample-splitting strategy, and establish its asymptotic selection consistency under some mild conditions. Effectiveness of the proposed selection criterion is demonstrated on a variety of numerical experiments and real-data examples.

Davy Paindaveine, Julien Remy, Thomas Verdebout.

Source: The Annals of Statistics, Volume 48, Number 1, 324--345.

Abstract:
We consider the problem of testing, on the basis of a $p$-variate Gaussian random sample, the null hypothesis $mathcal{H}_{0}:oldsymbol{ heta}_{1}=oldsymbol{ heta}_{1}^{0}$ against the alternative $mathcal{H}_{1}:oldsymbol{ heta}_{1} eq oldsymbol{ heta}_{1}^{0}$, where $oldsymbol{ heta}_{1}$ is the “first” eigenvector of the underlying covariance matrix and $oldsymbol{ heta}_{1}^{0}$ is a fixed unit $p$-vector. In the classical setup where eigenvalues $lambda_{1}>lambda_{2}geq cdots geq lambda_{p}$ are fixed, the Anderson ( Ann. Math. Stat. 34 (1963) 122–148) likelihood ratio test (LRT) and the Hallin, Paindaveine and Verdebout ( Ann. Statist. 38 (2010) 3245–3299) Le Cam optimal test for this problem are asymptotically equivalent under the null hypothesis, hence also under sequences of contiguous alternatives. We show that this equivalence does not survive asymptotic scenarios where $lambda_{n1}/lambda_{n2}=1+O(r_{n})$ with $r_{n}=O(1/sqrt{n})$. For such scenarios, the Le Cam optimal test still asymptotically meets the nominal level constraint, whereas the LRT severely overrejects the null hypothesis. Consequently, the former test should be favored over the latter one whenever the two largest sample eigenvalues are close to each other. By relying on the Le Cam’s asymptotic theory of statistical experiments, we study the non-null and optimality properties of the Le Cam optimal test in the aforementioned asymptotic scenarios and show that the null robustness of this test is not obtained at the expense of power. Our asymptotic investigation is extensive in the sense that it allows $r_{n}$ to converge to zero at an arbitrary rate. While we restrict to single-spiked spectra of the form $lambda_{n1}>lambda_{n2}=cdots =lambda_{np}$ to make our results as striking as possible, we extend our results to the more general elliptical case. Finally, we present an illustrative real data example.

Xiaoqin Wang, Li Yin.

Source: The Annals of Statistics, Volume 48, Number 1, 138--160.

Abstract:
In sequential causal inference, two types of causal effects are of practical interest, namely, the causal effect of the treatment regime (called the sequential causal effect) and the blip effect of treatment on the potential outcome after the last treatment. The well-known $G$-formula expresses these causal effects in terms of the standard parameters. In this article, we obtain a new $G$-formula that expresses these causal effects in terms of the point observable effects of treatments similar to treatment in the framework of single-point causal inference. Based on the new $G$-formula, we estimate these causal effects by maximum likelihood via point observable effects with methods extended from single-point causal inference. We are able to increase precision of the estimation without introducing biases by an unsaturated model imposing constraints on the point observable effects. We are also able to reduce the number of point observable effects in the estimation by treatment assignment conditions.

Emmanuel J. Candès, Pragya Sur.

Source: The Annals of Statistics, Volume 48, Number 1, 27--42.

Abstract:
This paper rigorously establishes that the existence of the maximum likelihood estimate (MLE) in high-dimensional logistic regression models with Gaussian covariates undergoes a sharp “phase transition.” We introduce an explicit boundary curve $h_{mathrm{MLE}}$, parameterized by two scalars measuring the overall magnitude of the unknown sequence of regression coefficients, with the following property: in the limit of large sample sizes $n$ and number of features $p$ proportioned in such a way that $p/n ightarrow kappa $, we show that if the problem is sufficiently high dimensional in the sense that $kappa >h_{mathrm{MLE}}$, then the MLE does not exist with probability one. Conversely, if $kappa <h_{mathrm{MLE}}$, the MLE asymptotically exists with probability one.

Francesco Bravo, Juan Carlos Escanciano, Ingrid Van Keilegom.

Source: The Annals of Statistics, Volume 48, Number 1, 1--26.

Abstract:
In both parametric and certain nonparametric statistical models, the empirical likelihood ratio satisfies a nonparametric version of Wilks’ theorem. For many semiparametric models, however, the commonly used two-step (plug-in) empirical likelihood ratio is not asymptotically distribution-free, that is, its asymptotic distribution contains unknown quantities, and hence Wilks’ theorem breaks down. This article suggests a general approach to restore Wilks’ phenomenon in two-step semiparametric empirical likelihood inferences. The main insight consists in using as the moment function in the estimating equation the influence function of the plug-in sample moment. The proposed method is general; it leads to a chi-squared limiting distribution with known degrees of freedom; it is efficient; it does not require undersmoothing; and it is less sensitive to the first-step than alternative methods, which is particularly appealing for high-dimensional settings. Several examples and simulation studies illustrate the general applicability of the procedure and its excellent finite sample performance relative to competing methods.

Zhigang Bao.

Source: The Annals of Statistics, Volume 47, Number 6, 3504--3532.

Abstract:
In this paper, we study a high-dimensional random matrix model from nonparametric statistics called the Kendall rank correlation matrix, which is a natural multivariate extension of the Kendall rank correlation coefficient. We establish the Tracy–Widom law for its largest eigenvalue. It is the first Tracy–Widom law for a nonparametric random matrix model, and also the first Tracy–Widom law for a high-dimensional U-statistic.

Monika Bhattacharjee, Arup Bose.

Source: The Annals of Statistics, Volume 47, Number 6, 3470--3503.

Abstract:
Consider a high-dimensional linear time series model where the dimension $p$ and the sample size $n$ grow in such a way that $p/n o 0$. Let $hat{Gamma }_{u}$ be the $u$th order sample autocovariance matrix. We first show that the LSD of any symmetric polynomial in ${hat{Gamma }_{u},hat{Gamma }_{u}^{*},ugeq 0}$ exists under independence and moment assumptions on the driving sequence together with weak assumptions on the coefficient matrices. This LSD result, with some additional effort, implies the asymptotic normality of the trace of any polynomial in ${hat{Gamma }_{u},hat{Gamma }_{u}^{*},ugeq 0}$. We also study similar results for several independent MA processes. We show applications of the above results to statistical inference problems such as in estimation of the unknown order of a high-dimensional MA process and in graphical and significance tests for hypotheses on coefficient matrices of one or several such independent processes.

Alessandro Rinaldo, Larry Wasserman, Max G’Sell.

Source: The Annals of Statistics, Volume 47, Number 6, 3438--3469.

Abstract:
Several new methods have been recently proposed for performing valid inference after model selection. An older method is sample splitting: use part of the data for model selection and the rest for inference. In this paper, we revisit sample splitting combined with the bootstrap (or the Normal approximation). We show that this leads to a simple, assumption-lean approach to inference and we establish results on the accuracy of the method. In fact, we find new bounds on the accuracy of the bootstrap and the Normal approximation for general nonlinear parameters with increasing dimension which we then use to assess the accuracy of regression inference. We define new parameters that measure variable importance and that can be inferred with greater accuracy than the usual regression coefficients. Finally, we elucidate an inference-prediction trade-off: splitting increases the accuracy and robustness of inference but can decrease the accuracy of the predictions.

Benjamin Eltzner, Stephan F. Huckemann.

Source: The Annals of Statistics, Volume 47, Number 6, 3360--3381.

Abstract:
The (CLT) central limit theorems for generalized Fréchet means (data descriptors assuming values in manifolds, such as intrinsic means, geodesics, etc.) on manifolds from the literature are only valid if a certain empirical process of Hessians of the Fréchet function converges suitably, as in the proof of the prototypical BP-CLT [ Ann. Statist. 33 (2005) 1225–1259]. This is not valid in many realistic scenarios and we provide for a new very general CLT. In particular, this includes scenarios where, in a suitable chart, the sample mean fluctuates asymptotically at a scale $n^{alpha }$ with exponents $alpha <1/2$ with a nonnormal distribution. As the BP-CLT yields only fluctuations that are, rescaled with $n^{1/2}$, asymptotically normal, just as the classical CLT for random vectors, these lower rates, somewhat loosely called smeariness, had to date been observed only on the circle. We make the concept of smeariness on manifolds precise, give an example for two-smeariness on spheres of arbitrary dimension, and show that smeariness, although “almost never” occurring, may have serious statistical implications on a continuum of sample scenarios nearby. In fact, this effect increases with dimension, striking in particular in high dimension low sample size scenarios.

Shurong Zheng, Zhao Chen, Hengjian Cui, Runze Li.

Source: The Annals of Statistics, Volume 47, Number 6, 3300--3334.

Abstract:
This paper is concerned with test of significance on high-dimensional covariance structures, and aims to develop a unified framework for testing commonly used linear covariance structures. We first construct a consistent estimator for parameters involved in the linear covariance structure, and then develop two tests for the linear covariance structures based on entropy loss and quadratic loss used for covariance matrix estimation. To study the asymptotic properties of the proposed tests, we study related high-dimensional random matrix theory, and establish several highly useful asymptotic results. With the aid of these asymptotic results, we derive the limiting distributions of these two tests under the null and alternative hypotheses. We further show that the quadratic loss based test is asymptotically unbiased. We conduct Monte Carlo simulation study to examine the finite sample performance of the two tests. Our simulation results show that the limiting null distributions approximate their null distributions quite well, and the corresponding asymptotic critical values keep Type I error rate very well. Our numerical comparison implies that the proposed tests outperform existing ones in terms of controlling Type I error rate and power. Our simulation indicates that the test based on quadratic loss seems to have better power than the test based on entropy loss.

Victor Veitch, Daniel M. Roy.

Source: The Annals of Statistics, Volume 47, Number 6, 3274--3299.

Abstract:
Sparse exchangeable graphs on $mathbb{R}_{+}$, and the associated graphex framework for sparse graphs, generalize exchangeable graphs on $mathbb{N}$, and the associated graphon framework for dense graphs. We develop the graphex framework as a tool for statistical network analysis by identifying the sampling scheme that is naturally associated with the models of the framework, formalizing two natural notions of consistent estimation of the parameter (the graphex) underlying these models, and identifying general consistent estimators in each case. The sampling scheme is a modification of independent vertex sampling that throws away vertices that are isolated in the sampled subgraph. The estimators are variants of the empirical graphon estimator, which is known to be a consistent estimator for the distribution of dense exchangeable graphs; both can be understood as graph analogues to the empirical distribution in the i.i.d. sequence setting. Our results may be viewed as a generalization of consistent estimation via the empirical graphon from the dense graph regime to also include sparse graphs.

Long Feng, Cun-Hui Zhang.

Source: The Annals of Statistics, Volume 47, Number 6, 3069--3098.

Abstract:
The Lasso is biased. Concave penalized least squares estimation (PLSE) takes advantage of signal strength to reduce this bias, leading to sharper error bounds in prediction, coefficient estimation and variable selection. For prediction and estimation, the bias of the Lasso can be also reduced by taking a smaller penalty level than what selection consistency requires, but such smaller penalty level depends on the sparsity of the true coefficient vector. The sorted $ell_{1}$ penalized estimation (Slope) was proposed for adaptation to such smaller penalty levels. However, the advantages of concave PLSE and Slope do not subsume each other. We propose sorted concave penalized estimation to combine the advantages of concave and sorted penalizations. We prove that sorted concave penalties adaptively choose the smaller penalty level and at the same time benefits from signal strength, especially when a significant proportion of signals are stronger than the corresponding adaptively selected penalty levels. A local convex approximation for sorted concave penalties, which extends the local linear and quadratic approximations for separable concave penalties, is developed to facilitate the computation of sorted concave PLSE and proven to possess desired prediction and estimation error bounds. Our analysis of prediction and estimation errors requires the restricted eigenvalue condition on the design, not beyond, and provides selection consistency under a required minimum signal strength condition in addition. Thus, our results also sharpens existing results on concave PLSE by removing the upper sparse eigenvalue component of the sparse Riesz condition.

Heng Lian, Kaifeng Zhao, Shaogao Lv.

Source: The Annals of Statistics, Volume 47, Number 5, 2922--2949.

Abstract:
In this paper, we consider the local asymptotics of the nonparametric function in a partially linear model, within the framework of the divide-and-conquer estimation. Unlike the fixed-dimensional setting in which the parametric part does not affect the nonparametric part, the high-dimensional setting makes the issue more complicated. In particular, when a sparsity-inducing penalty such as lasso is used to make the estimation of the linear part feasible, the bias introduced will propagate to the nonparametric part. We propose a novel approach for estimation of the nonparametric function and establish the local asymptotics of the estimator. The result is useful for massive data with possibly different linear coefficients in each subpopulation but common nonparametric function. Some numerical illustrations are also presented.

Chengchun Shi, Rui Song, Zhao Chen, Runze Li.

Source: The Annals of Statistics, Volume 47, Number 5, 2671--2703.

Abstract:
This paper is concerned with testing linear hypotheses in high dimensional generalized linear models. To deal with linear hypotheses, we first propose the constrained partial regularization method and study its statistical properties. We further introduce an algorithm for solving regularization problems with folded-concave penalty functions and linear constraints. To test linear hypotheses, we propose a partial penalized likelihood ratio test, a partial penalized score test and a partial penalized Wald test. We show that the limiting null distributions of these three test statistics are $chi^{2}$ distribution with the same degrees of freedom, and under local alternatives, they asymptotically follow noncentral $chi^{2}$ distributions with the same degrees of freedom and noncentral parameter, provided the number of parameters involved in the test hypothesis grows to $infty$ at a certain rate. Simulation studies are conducted to examine the finite sample performance of the proposed tests. Empirical analysis of a real data example is used to illustrate the proposed testing procedures.

Zhiqiang Tan, Cun-Hui Zhang.

Source: The Annals of Statistics, Volume 47, Number 5, 2567--2600.

Abstract:
Additive regression provides an extension of linear regression by modeling the signal of a response as a sum of functions of covariates of relatively low complexity. We study penalized estimation in high-dimensional nonparametric additive regression where functional semi-norms are used to induce smoothness of component functions and the empirical $L_{2}$ norm is used to induce sparsity. The functional semi-norms can be of Sobolev or bounded variation types and are allowed to be different amongst individual component functions. We establish oracle inequalities for the predictive performance of such methods under three simple technical conditions: a sub-Gaussian condition on the noise, a compatibility condition on the design and the functional classes under consideration and an entropy condition on the functional classes. For random designs, the sample compatibility condition can be replaced by its population version under an additional condition to ensure suitable convergence of empirical norms. In homogeneous settings where the complexities of the component functions are of the same order, our results provide a spectrum of minimax convergence rates, from the so-called slow rate without requiring the compatibility condition to the fast rate under the hard sparsity or certain $L_{q}$ sparsity to allow many small components in the true regression function. These results significantly broaden and sharpen existing ones in the literature.

Joshua Cape, Minh Tang, Carey E. Priebe.

Source: The Annals of Statistics, Volume 47, Number 5, 2405--2439.

Abstract:
The singular value matrix decomposition plays a ubiquitous role throughout statistics and related fields. Myriad applications including clustering, classification, and dimensionality reduction involve studying and exploiting the geometric structure of singular values and singular vectors. This paper provides a novel collection of technical and theoretical tools for studying the geometry of singular subspaces using the two-to-infinity norm. Motivated by preliminary deterministic Procrustes analysis, we consider a general matrix perturbation setting in which we derive a new Procrustean matrix decomposition. Together with flexible machinery developed for the two-to-infinity norm, this allows us to conduct a refined analysis of the induced perturbation geometry with respect to the underlying singular vectors even in the presence of singular value multiplicity. Our analysis yields singular vector entrywise perturbation bounds for a range of popular matrix noise models, each of which has a meaningful associated statistical inference task. In addition, we demonstrate how the two-to-infinity norm is the preferred norm in certain statistical settings. Specific applications discussed in this paper include covariance estimation, singular subspace recovery, and multiple graph inference. Both our Procrustean matrix decomposition and the technical machinery developed for the two-to-infinity norm may be of independent interest.

Stefan Richter, Rainer Dahlhaus.

Source: The Annals of Statistics, Volume 47, Number 4, 2145--2173.

Abstract:
We propose an adaptive bandwidth selector via cross validation for local M-estimators in locally stationary processes. We prove asymptotic optimality of the procedure under mild conditions on the underlying parameter curves. The results are applicable to a wide range of locally stationary processes such linear and nonlinear processes. A simulation study shows that the method works fairly well also in misspecified situations.

Chengchun Shi, Rui Song, Wenbin Lu.

Source: The Annals of Statistics, Volume 47, Number 4, 2348--2377.

Abstract:
Precision medicine is an emerging medical paradigm that focuses on finding the most effective treatment strategy tailored for individual patients. In the literature, most of the existing works focused on estimating the optimal treatment regime. However, there has been less attention devoted to hypothesis testing regarding the optimal treatment regime. In this paper, we first introduce the notion of conditional qualitative treatment effects (CQTE) of a set of variables given another set of variables and provide a class of equivalent representations for the null hypothesis of no CQTE. The proposed definition of CQTE does not assume any parametric form for the optimal treatment rule and plays an important role for assessing the incremental value of a set of new variables in optimal treatment decision making conditional on an existing set of prescriptive variables. We then propose novel testing procedures for no CQTE based on kernel estimation of the conditional contrast functions. We show that our test statistics have asymptotically correct size and nonnegligible power against some nonstandard local alternatives. The empirical performance of the proposed tests are evaluated by simulations and an application to an AIDS data set.

Benedikt Bauer, Michael Kohler.

Source: The Annals of Statistics, Volume 47, Number 4, 2261--2285.

Abstract:
Assuming that a smoothness condition and a suitable restriction on the structure of the regression function hold, it is shown that least squares estimates based on multilayer feedforward neural networks are able to circumvent the curse of dimensionality in nonparametric regression. The proof is based on new approximation results concerning multilayer feedforward neural networks with bounded weights and a bounded number of hidden neurons. The estimates are compared with various other approaches by using simulated data.

Mathieu Gerber, Nicolas Chopin, Nick Whiteley.

Source: The Annals of Statistics, Volume 47, Number 4, 2236--2260.

Abstract:
We study convergence and convergence rates for resampling schemes. Our first main result is a general consistency theorem based on the notion of negative association, which is applied to establish the almost sure weak convergence of measures output from Kitagawa’s [ J. Comput. Graph. Statist. 5 (1996) 1–25] stratified resampling method. Carpenter, Ckiffird and Fearnhead’s [ IEE Proc. Radar Sonar Navig. 146 (1999) 2–7] systematic resampling method is similar in structure but can fail to converge depending on the order of the input samples. We introduce a new resampling algorithm based on a stochastic rounding technique of [In 42nd IEEE Symposium on Foundations of Computer Science ( Las Vegas , NV , 2001) (2001) 588–597 IEEE Computer Soc.], which shares some attractive properties of systematic resampling, but which exhibits negative association and, therefore, converges irrespective of the order of the input samples. We confirm a conjecture made by [ J. Comput. Graph. Statist. 5 (1996) 1–25] that ordering input samples by their states in $mathbb{R}$ yields a faster rate of convergence; we establish that when particles are ordered using the Hilbert curve in $mathbb{R}^{d}$, the variance of the resampling error is ${scriptstylemathcal{O}}(N^{-(1+1/d)})$ under mild conditions, where $N$ is the number of particles. We use these results to establish asymptotic properties of particle algorithms based on resampling schemes that differ from multinomial resampling.

Yen-Chi Chen.

Source: The Annals of Statistics, Volume 47, Number 4, 2174--2203.

Abstract:
In this paper we study the $alpha $-cluster tree ($alpha $-tree) under both singular and nonsingular measures. The $alpha $-tree uses probability contents within a set created by the ordering of points to construct a cluster tree so that it is well defined even for singular measures. We first derive the convergence rate for a density level set around critical points, which leads to the convergence rate for estimating an $alpha $-tree under nonsingular measures. For singular measures, we study how the kernel density estimator (KDE) behaves and prove that the KDE is not uniformly consistent but pointwise consistent after rescaling. We further prove that the estimated $alpha $-tree fails to converge in the $L_{infty }$ metric but is still consistent under the integrated distance. We also observe a new type of critical points—the dimensional critical points (DCPs)—of a singular measure. DCPs are points that contribute to cluster tree topology but cannot be defined using density gradient. Building on the analysis of the KDE and DCPs, we prove the topological consistency of an estimated $alpha $-tree.

James G. Scott, James O. Berger

Source: Ann. Statist., Volume 38, Number 5, 2587--2619.

Abstract:
This paper studies the multiplicity-correction effect of standard Bayesian variable-selection priors in linear regression. Our first goal is to clarify when, and how, multiplicity correction happens automatically in Bayesian analysis, and to distinguish this correction from the Bayesian Ockham’s-razor effect. Our second goal is to contrast empirical-Bayes and fully Bayesian approaches to variable selection through examples, theoretical results and simulations. Considerable differences between the two approaches are found. In particular, we prove a theorem that characterizes a surprising aymptotic discrepancy between fully Bayes and empirical Bayes. This discrepancy arises from a different source than the failure to account for hyperparameter uncertainty in the empirical-Bayes estimate. Indeed, even at the extreme, when the empirical-Bayes estimate converges asymptotically to the true variable-inclusion probability, the potential for a serious difference remains.

Ability to work with each other. In the loosely coupled environment of a service-oriented architecture, separate resources don't need to know the details of how they each work, but they need to have enough common ground to reliably exchange messages without error or misunderstanding. Standardized specifications go a long way towards creating this common ground, but differences in implementation may still lead to breakdowns in communication. Interoperability is when services can interact with each other without encountering such problems.

Digital identity standards group. Set up at the instigation of Sun Microsystems in 2001, the Liberty Alliance Project is a consortium of technology vendors and consumer-facing enterprises formed "to establish an open standard for federated network identity." It aims to make it easier for consumers to access networked services from multiple suppliers while safeguarding security and privacy. Its specifications have been published in three phases: the Identity Federation Framework (ID-FF) came first; the Identity Web Services Framework (ID-WSF) followed in November 2003; and work is in progress on the Identity Services Interface Specifications (ID-SIS). Liberty Alliance specifications are closely linked to the SAML single sign-on standard, and overlap with elements of WS-Security.

Trang Quynh Nguyen, Elizabeth A. Stuart.

Source: The Annals of Applied Statistics, Volume 14, Number 1, 518--520.

Alex Diana, Eleni Matechou, Jim Griffin, Alison Johnston.

Source: The Annals of Applied Statistics, Volume 14, Number 1, 473--493.

Abstract:
Environmental changes in recent years have been linked to phenological shifts which in turn are linked to the survival of species. The work in this paper is motivated by capture-recapture data on blackcaps collected by the British Trust for Ornithology as part of the Constant Effort Sites monitoring scheme. Blackcaps overwinter abroad and migrate to the UK annually for breeding purposes. We propose a novel Bayesian nonparametric approach for expressing the bivariate density of individual arrival and departure times at different sites across a number of years as a mixture model. The new model combines the ideas of the hierarchical and the dependent Dirichlet process, allowing the estimation of site-specific weights and year-specific mixture locations, which are modelled as functions of environmental covariates using a multivariate extension of the Gaussian process. The proposed modelling framework is extremely general and can be used in any context where multivariate density estimation is performed jointly across different groups and in the presence of a continuous covariate.

Yicheng Li, Adrian E. Raftery.

Source: The Annals of Applied Statistics, Volume 14, Number 1, 381--408.

Abstract:
Smoking is one of the leading preventable threats to human health and a major risk factor for lung cancer, upper aerodigestive cancer and chronic obstructive pulmonary disease. Estimating and forecasting the smoking attributable fraction (SAF) of mortality can yield insights into smoking epidemics and also provide a basis for more accurate mortality and life expectancy projection. Peto et al. ( Lancet 339 (1992) 1268–1278) proposed a method to estimate the SAF using the lung cancer mortality rate as an indicator of exposure to smoking in the population of interest. Here, we use the same method to estimate the all-age SAF (ASAF) for both genders for over 60 countries. We document a strong and cross-nationally consistent pattern of the evolution of the SAF over time. We use this as the basis for a new Bayesian hierarchical model to project future male and female ASAF from over 60 countries simultaneously. This gives forecasts as well as predictive distributions that can be used to find uncertainty intervals for any quantity of interest. We assess the model using out-of-sample predictive validation and find that it provides good forecasts and well-calibrated forecast intervals, comparing favorably with other methods.

Peng Shi, Zifeng Zhao.

Source: The Annals of Applied Statistics, Volume 14, Number 1, 357--380.

Abstract:
In actuarial research a task of particular interest and importance is to predict the loss cost for individual risks so that informative decisions are made in various insurance operations such as underwriting, ratemaking and capital management. The loss cost is typically viewed to follow a compound distribution where the summation of the severity variables is stopped by the frequency variable. A challenging issue in modeling such outcomes is to accommodate the potential dependence between the number of claims and the size of each individual claim. In this article we introduce a novel regression framework for compound distributions that uses a copula to accommodate the association between the frequency and the severity variables and, thus, allows for arbitrary dependence between the two components. We further show that the new model is very flexible and is easily modified to account for incomplete data due to censoring or truncation. The flexibility of the proposed model is illustrated using both simulated and real data sets. In the analysis of granular claims data from property insurance, we find substantive negative relationship between the number and the size of insurance claims. In addition, we demonstrate that ignoring the frequency-severity association could lead to biased decision-making in insurance operations.

Medha Uppala, Mark S. Handcock.

Source: The Annals of Applied Statistics, Volume 14, Number 1, 339--356.

Abstract:
This paper focuses on spatial and temporal modeling of point processes on linear networks. Point processes on linear networks can simply be defined as point events occurring on or near line segment network structures embedded in a certain space. A separable modeling framework is introduced that posits separate formation and dissolution models of point processes on linear networks over time. While the model was inspired by spider web building activity in brick mortar lines, the focus is on modeling wildfire ignition origins near road networks over a span of 14 years. As most wildfires in California have human-related origins, modeling the origin locations with respect to the road network provides insight into how human, vehicular and structural densities affect ignition occurrence. Model results show that roads that traverse different types of regions such as residential, interface and wildland regions have higher ignition intensities compared to roads that only exist in each of the mentioned region types.

Jared D. Fisher, Davide Pettenuzzo, Carlos M. Carvalho.

Source: The Annals of Applied Statistics, Volume 14, Number 1, 299--338.

Abstract:
We introduce a fast, closed-form, simulation-free method to model and forecast multiple asset returns and employ it to investigate the optimal ensemble of features to include when jointly predicting monthly stock and bond excess returns. Our approach builds on the Bayesian dynamic linear models of West and Harrison ( Bayesian Forecasting and Dynamic Models (1997) Springer), and it can objectively determine, through a fully automated procedure, both the optimal set of regressors to include in the predictive system and the degree to which the model coefficients, volatilities and covariances should vary over time. When applied to a portfolio of five stock and bond returns, we find that our method leads to large forecast gains, both in statistical and economic terms. In particular, we find that relative to a standard no-predictability benchmark, the optimal combination of predictors, stochastic volatility and time-varying covariances increases the annualized certainty equivalent returns of a leverage-constrained power utility investor by more than 500 basis points.

Wanghuan Chu, Runze Li, Jingyuan Liu, Matthew Reimherr.

Source: The Annals of Applied Statistics, Volume 14, Number 1, 276--298.

Abstract:
Motivated by an empirical analysis of data from a genome-wide association study on obesity, measured by the body mass index (BMI), we propose a two-step gene-detection procedure for generalized varying coefficient mixed-effects models with ultrahigh dimensional covariates. The proposed procedure selects significant single nucleotide polymorphisms (SNPs) impacting the mean BMI trend, some of which have already been biologically proven to be “fat genes.” The method also discovers SNPs that significantly influence the age-dependent variability of BMI. The proposed procedure takes into account individual variations of genetic effects and can also be directly applied to longitudinal data with continuous, binary or count responses. We employ Monte Carlo simulation studies to assess the performance of the proposed method and further carry out causal inference for the selected SNPs.

Tsuyoshi Kunihama, Zehang Richard Li, Samuel J. Clark, Tyler H. McCormick.

Source: The Annals of Applied Statistics, Volume 14, Number 1, 241--256.

Abstract:
The distribution of deaths by cause provides crucial information for public health planning, response and evaluation. About 60% of deaths globally are not registered or given a cause, limiting our ability to understand disease epidemiology. Verbal autopsy (VA) surveys are increasingly used in such settings to collect information on the signs, symptoms and medical history of people who have recently died. This article develops a novel Bayesian method for estimation of population distributions of deaths by cause using verbal autopsy data. The proposed approach is based on a multivariate probit model where associations among items in questionnaires are flexibly induced by latent factors. Using the Population Health Metrics Research Consortium labeled data that include both VA and medically certified causes of death, we assess performance of the proposed method. Further, we estimate important questionnaire items that are highly associated with causes of death. This framework provides insights that will simplify future data

Zhenguo Gao, Pang Du, Ran Jin, John L. Robertson.

Source: The Annals of Applied Statistics, Volume 14, Number 1, 143--159.

Abstract:
Liver procurement experiments with surface-temperature monitoring motivated Gao et al. ( J. Amer. Statist. Assoc. 114 (2019) 773–781) to develop a variance change-point detection method under a smoothly-changing mean trend. However, the spotwise change points yielded from their method do not offer immediate information to surgeons since an organ is often transplanted as a whole or in part. We develop a new practical method that can analyze a defined portion of the organ surface at a time. It also provides a novel addition to the developing field of functional data monitoring. Furthermore, numerical challenge emerges for simultaneously modeling the variance functions of 2D locations and the mean function of location and time. The respective sample sizes in the scales of 10,000 and 1,000,000 for modeling these functions make standard spline estimation too costly to be useful. We introduce a multistage subsampling strategy with steps educated by quickly-computable preliminary statistical measures. Extensive simulations show that the new method can efficiently reduce the computational cost and provide reasonable parameter estimates. Application of the new method to our liver surface temperature monitoring data shows its effectiveness in providing accurate status change information for a selected portion of the organ in the experiment.

Bryan D. Martin, Daniela Witten, Amy D. Willis.

Source: The Annals of Applied Statistics, Volume 14, Number 1, 94--115.

Abstract:
Using a sample from a population to estimate the proportion of the population with a certain category label is a broadly important problem. In the context of microbiome studies, this problem arises when researchers wish to use a sample from a population of microbes to estimate the population proportion of a particular taxon, known as the taxon’s relative abundance . In this paper, we propose a beta-binomial model for this task. Like existing models, our model allows for a taxon’s relative abundance to be associated with covariates of interest. However, unlike existing models, our proposal also allows for the overdispersion in the taxon’s counts to be associated with covariates of interest. We exploit this model in order to propose tests not only for differential relative abundance, but also for differential variability. The latter is particularly valuable in light of speculation that dysbiosis , the perturbation from a normal microbiome that can occur in certain disease conditions, may manifest as a loss of stability, or increase in variability, of the counts associated with each taxon. We demonstrate the performance of our proposed model using a simulation study and an application to soil microbial data.

Wenjie Wang, Robert Aseltine, Kun Chen, Jun Yan.

Source: The Annals of Applied Statistics, Volume 14, Number 1, 51--73.

Abstract:
The concept of integrating data from disparate sources to accelerate scientific discovery has generated tremendous excitement in many fields. The potential benefits from data integration, however, may be compromised by the uncertainty due to incomplete/imperfect record linkage. Motivated by a suicide risk study, we propose an approach for analyzing survival data with uncertain event times arising from data integration. Specifically, in our problem deaths identified from the hospital discharge records together with reported suicidal deaths determined by the Office of Medical Examiner may still not include all the death events of patients, and the missing deaths can be recovered from a complete database of death records. Since the hospital discharge data can only be linked to the death record data by matching basic patient characteristics, a patient with a censored death time from the first dataset could be linked to multiple potential event records in the second dataset. We develop an integrative Cox proportional hazards regression in which the uncertainty in the matched event times is modeled probabilistically. The estimation procedure combines the ideas of profile likelihood and the expectation conditional maximization algorithm (ECM). Simulation studies demonstrate that under realistic settings of imperfect data linkage the proposed method outperforms several competing approaches including multiple imputation. A marginal screening analysis using the proposed integrative Cox model is performed to identify risk factors associated with death following suicide-related hospitalization in Connecticut. The identified diagnostics codes are consistent with existing literature and provide several new insights on suicide risk, prediction and prevention.

Francisco J. R. Ruiz, Susan Athey, David M. Blei.

Source: The Annals of Applied Statistics, Volume 14, Number 1, 1--27.

Abstract:
We develop SHOPPER, a sequential probabilistic model of shopping data. SHOPPER uses interpretable components to model the forces that drive how a customer chooses products; in particular, we designed SHOPPER to capture how items interact with other items. We develop an efficient posterior inference algorithm to estimate these forces from large-scale data, and we analyze a large dataset from a major chain grocery store. We are interested in answering counterfactual queries about changes in prices. We found that SHOPPER provides accurate predictions even under price interventions, and that it helps identify complementary and substitutable pairs of products.

Joe Watson, James V. Zidek, Gavin Shaddick.

Source: The Annals of Applied Statistics, Volume 13, Number 4, 2662--2700.

Abstract:
This paper presents a general model framework for detecting the preferential sampling of environmental monitors recording an environmental process across space and/or time. This is achieved by considering the joint distribution of an environmental process with a site-selection process that considers where and when sites are placed to measure the process. The environmental process may be spatial, temporal or spatio-temporal in nature. By sharing random effects between the two processes, the joint model is able to establish whether site placement was stochastically dependent of the environmental process under study. Furthermore, if stochastic dependence is identified between the two processes, then inferences about the probability distribution of the spatio-temporal process will change, as will predictions made of the process across space and time. The embedding into a spatio-temporal framework also allows for the modelling of the dynamic site-selection process itself. Real-world factors affecting both the size and location of the network can be easily modelled and quantified. Depending upon the choice of the population of locations considered for selection across space and time under the site-selection process, different insights about the precise nature of preferential sampling can be obtained. The general framework developed in the paper is designed to be easily and quickly fit using the R-INLA package. We apply this framework to a case study involving particulate air pollution over the UK where a major reduction in the size of a monitoring network through time occurred. It is demonstrated that a significant response-biased reduction in the air quality monitoring network occurred, namely the relocation of monitoring sites to locations with the highest pollution levels, and the routine removal of sites at locations with the lowest. We also show that the network was consistently unrepresenting levels of particulate matter seen across much of GB throughout the operating life of the network. Finally we show that this may have led to a severe overreporting of the population-average exposure levels experienced across GB. This could have great impacts on estimates of the health effects of black smoke levels.

Elizabeth Lorenzi, Ricardo Henao, Katherine Heller.

Source: The Annals of Applied Statistics, Volume 13, Number 4, 2637--2661.

Abstract:
Nearly a third of all surgeries performed in the United States occur for patients over the age of 65; these older adults experience a higher rate of postoperative morbidity and mortality. To improve the care for these patients, we aim to identify and characterize high risk geriatric patients to send to a specialized perioperative clinic while leveraging the overall surgical population to improve learning. To this end, we develop a hierarchical infinite latent factor model (HIFM) to appropriately account for the covariance structure across subpopulations in data. We propose a novel Hierarchical Dirichlet Process shrinkage prior on the loadings matrix that flexibly captures the underlying structure of our data while sharing information across subpopulations to improve inference and prediction. The stick-breaking construction of the prior assumes an infinite number of factors and allows for each subpopulation to utilize different subsets of the factor space and select the number of factors needed to best explain the variation. We develop the model into a latent factor regression method that excels at prediction and inference of regression coefficients. Simulations validate this strong performance compared to baseline methods. We apply this work to the problem of predicting surgical complications using electronic health record data for geriatric patients and all surgical patients at Duke University Health System (DUHS). The motivating application demonstrates the improved predictive performance when using HIFM in both area under the ROC curve and area under the PR Curve while providing interpretable coefficients that may lead to actionable interventions.

Li Zhu, Zhiguang Huo, Tianzhou Ma, Steffi Oesterreich, George C. Tseng.

Source: The Annals of Applied Statistics, Volume 13, Number 4, 2611--2636.

Abstract:
Variable selection is a pervasive problem in modern high-dimensional data analysis where the number of features often exceeds the sample size (a.k.a. small-n-large-p problem). Incorporation of group structure knowledge to improve variable selection has been widely studied. Here, we consider prior knowledge of a hierarchical overlapping group structure to improve variable selection in regression setting. In genomics applications, for instance, a biological pathway contains tens to hundreds of genes and a gene can be mapped to multiple experimentally measured features (such as its mRNA expression, copy number variation and methylation levels of possibly multiple sites). In addition to the hierarchical structure, the groups at the same level may overlap (e.g., two pathways can share common genes). Incorporating such hierarchical overlapping groups in traditional penalized regression setting remains a difficult optimization problem. Alternatively, we propose a Bayesian indicator model that can elegantly serve the purpose. We evaluate the model in simulations and two breast cancer examples, and demonstrate its superior performance over existing models. The result not only enhances prediction accuracy but also improves variable selection and model interpretation that lead to deeper biological insight of the disease.

Armin Schwartzman, Andrew J. Schork, Rong Zablocki, Wesley K. Thompson.

Source: The Annals of Applied Statistics, Volume 13, Number 4, 2509--2538.

Abstract:
Analysis of genome-wide association studies (GWAS) is characterized by a large number of univariate regressions where a quantitative trait is regressed on hundreds of thousands to millions of single-nucleotide polymorphism (SNP) allele counts, one at a time. This article proposes an estimator of the SNP heritability of the trait, defined here as the fraction of the variance of the trait explained by the SNPs in the study. The proposed GWAS heritability (GWASH) estimator is easy to compute, highly interpretable and is consistent as the number of SNPs and the sample size increase. More importantly, it can be computed from summary statistics typically reported in GWAS, not requiring access to the original data. The estimator takes full account of the linkage disequilibrium (LD) or correlation between the SNPs in the study through moments of the LD matrix, estimable from auxiliary datasets. Unlike other proposed estimators in the literature, we establish the theoretical properties of the GWASH estimator and obtain analytical estimates of the precision, allowing for power and sample size calculations for SNP heritability estimates and forming a firm foundation for future methodological development.

Kun Liang.

Source: The Annals of Applied Statistics, Volume 13, Number 4, 2452--2482.

Abstract:
Finding differentially expressed genes is a common task in high-throughput transcriptome studies. While traditional statistical methods rank the genes by their test statistics alone, we analyze an RNA sequencing dataset using the auxiliary information of gene length and the test statistics from a related microarray study. Given the auxiliary information, we propose a novel nonparametric empirical Bayes procedure to estimate the posterior probability of differential expression for each gene. We demonstrate the advantage of our procedure in extensive simulation studies and a psoriasis RNA sequencing study. The companion R package calm is available at Bioconductor.

Dale L. Zimmerman, Jun Tang, Rui Huang.

Source: The Annals of Applied Statistics, Volume 13, Number 4, 2416--2451.

Abstract:
We extend statistical shape analytic methods known as outline analysis for application to the strike zone, a central feature of the game of baseball. Although the strike zone is rigorously defined by Major League Baseball’s official rules, umpires make mistakes in calling pitches as strikes (and balls) and may even adhere to a strike zone somewhat different than that prescribed by the rule book. Our methods yield inference on geometric attributes (centroid, dimensions, orientation and shape) of this “called strike zone” (CSZ) and on the effects that years, umpires, player attributes, game situation factors and their interactions have on those attributes. The methodology consists of first using kernel discriminant analysis to determine a noisy outline representing the CSZ corresponding to each factor combination, then fitting existing elliptic Fourier and new generalized superelliptic models for closed curves to that outline and finally analyzing the fitted model coefficients using standard methods of regression analysis, factorial analysis of variance and variance component estimation. We apply these methods to PITCHf/x data comprising more than three million called pitches from the 2008–2016 Major League Baseball seasons to address numerous questions about the CSZ. We find that all geometric attributes of the CSZ, except its size, became significantly more like those of the rule-book strike zone from 2008–2016 and that several player attribute/game situation factors had statistically and practically significant effects on many of them. We also establish that the variation in the horizontal center, width and area of an individual umpire’s CSZ from pitch to pitch is smaller than their variation among CSZs from different umpires.

John R. Tipton, Mevin B. Hooten, Connor Nolan, Robert K. Booth, Jason McLachlan.

Source: The Annals of Applied Statistics, Volume 13, Number 4, 2363--2388.

Abstract:
Multivariate compositional count data arise in many applications including ecology, microbiology, genetics and paleoclimate. A frequent question in the analysis of multivariate compositional count data is what underlying values of a covariate(s) give rise to the observed composition. Learning the relationship between covariates and the compositional count allows for inverse prediction of unobserved covariates given compositional count observations. Gaussian processes provide a flexible framework for modeling functional responses with respect to a covariate without assuming a functional form. Many scientific disciplines use Gaussian process approximations to improve prediction and make inference on latent processes and parameters. When prediction is desired on unobserved covariates given realizations of the response variable, this is called inverse prediction. Because inverse prediction is often mathematically and computationally challenging, predicting unobserved covariates often requires fitting models that are different from the hypothesized generative model. We present a novel computational framework that allows for efficient inverse prediction using a Gaussian process approximation to generative models. Our framework enables scientific learning about how the latent processes co-vary with respect to covariates while simultaneously providing predictions of missing covariates. The proposed framework is capable of efficiently exploring the high dimensional, multi-modal latent spaces that arise in the inverse problem. To demonstrate flexibility, we apply our method in a generalized linear model framework to predict latent climate states given multivariate count data. Based on cross-validation, our model has predictive skill competitive with current methods while simultaneously providing formal, statistical inference on the underlying community dynamics of the biological system previously not available.

Federico Castelletti, Guido Consonni.

Source: The Annals of Applied Statistics, Volume 13, Number 4, 2289--2311.

Abstract:
A signalling pathway is a sequence of chemical reactions initiated by a stimulus which in turn affects a receptor, and then through some intermediate steps cascades down to the final cell response. Based on the technique of flow cytometry, samples of cell-by-cell measurements are collected under each experimental condition, resulting in a collection of interventional data (assuming no latent variables are involved). Usually several external interventions are applied at different points of the pathway, the ultimate aim being the structural recovery of the underlying signalling network which we model as a causal Directed Acyclic Graph (DAG) using intervention calculus. The advantage of using interventional data, rather than purely observational one, is that identifiability of the true data generating DAG is enhanced. More technically a Markov equivalence class of DAGs, whose members are statistically indistinguishable based on observational data alone, can be further decomposed, using additional interventional data, into smaller distinct Interventional Markov equivalence classes. We present a Bayesian methodology for structural learning of Interventional Markov equivalence classes based on observational and interventional samples of multivariate Gaussian observations. Our approach is objective, meaning that it is based on default parameter priors requiring no personal elicitation; some flexibility is however allowed through a tuning parameter which regulates sparsity in the prior on model space. Based on an analytical expression for the marginal likelihood of a given Interventional Essential Graph, and a suitable MCMC scheme, our analysis produces an approximate posterior distribution on the space of Interventional Markov equivalence classes, which can be used to provide uncertainty quantification for features of substantive scientific interest, such as the posterior probability of inclusion of selected edges, or paths.

Cecilia Balocchi, Shane T. Jensen.

Source: The Annals of Applied Statistics, Volume 13, Number 4, 2235--2259.

Abstract:
Understanding the relationship between change in crime over time and the geography of urban areas is an important problem for urban planning. Accurate estimation of changing crime rates throughout a city would aid law enforcement as well as enable studies of the association between crime and the built environment. Bayesian modeling is a promising direction since areal data require principled sharing of information to address spatial autocorrelation between proximal neighborhoods. We develop several Bayesian approaches to spatial sharing of information between neighborhoods while modeling trends in crime counts over time. We apply our methodology to estimate changes in crime throughout Philadelphia over the 2006-15 period while also incorporating spatially-varying economic and demographic predictors. We find that the local shrinkage imposed by a conditional autoregressive model has substantial benefits in terms of out-of-sample predictive accuracy of crime. We also explore the possibility of spatial discontinuities between neighborhoods that could represent natural barriers or aspects of the built environment.

Carolyn M. Rutter, Jonathan Ozik, Maria DeYoreo, Nicholson Collier.

Source: The Annals of Applied Statistics, Volume 13, Number 4, 2189--2212.

Abstract:
Microsimulation models (MSMs) are used to inform policy by predicting population-level outcomes under different scenarios. MSMs simulate individual-level event histories that mark the disease process (such as the development of cancer) and the effect of policy actions (such as screening) on these events. MSMs often have many unknown parameters; calibration is the process of searching the parameter space to select parameters that result in accurate MSM prediction of a wide range of targets. We develop Incremental Mixture Approximate Bayesian Computation (IMABC) for MSM calibration which results in a simulated sample from the posterior distribution of model parameters given calibration targets. IMABC begins with a rejection-based ABC step, drawing a sample of points from the prior distribution of model parameters and accepting points that result in simulated targets that are near observed targets. Next, the sample is iteratively updated by drawing additional points from a mixture of multivariate normal distributions and accepting points that result in accurate predictions. Posterior estimates are obtained by weighting the final set of accepted points to account for the adaptive sampling scheme. We demonstrate IMABC by calibrating CRC-SPIN 2.0, an updated version of a MSM for colorectal cancer (CRC) that has been used to inform national CRC screening guidelines.