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Xiao-Li Meng

Publications and source records attributed to Xiao-Li Meng.

At least 37 records · Page 2Linked to original sources
arXiv2020

Congenial Differential Privacy under Mandated Disclosure

Differentially private data releases are often required to satisfy a set of external constraints that reflect the legal, ethical, and logical mandates to which the data curator is obligated. The enforcement of constraints, when treated as post-processing, adds an extra phase in the production of privatized data. It is well understood in the theory of multi-phase processing that congeniality, a form of procedural compatibility between phases, is a prerequisite for the end users to straightforwardly obtain statistically valid results. Congenial differential privacy is theoretically principled, which facilitates transparency and intelligibility of the mechanism that would otherwise be undermined by ad-hoc post-processing procedures. We advocate for the systematic integration of mandated disclosure into the design of the privacy mechanism via standard probabilistic conditioning on the invariant margins. Conditioning automatically renders congeniality because any extra post-processing phase becomes unnecessary. We provide both initial theoretical guarantees and a Markov chain algorithm for our proposal. We also discuss intriguing theoretical issues that arise in comparing congenital differential privacy and optimization-based post-processing, as well as directions for further research.

cs.CR↗
arXiv2019

Designing Test Information and Test Information in Design

DeGroot (1962) developed a general framework for constructing Bayesian measures of the expected information that an experiment will provide for estimation. We propose an analogous framework for measures of information for hypothesis testing. In contrast to estimation information measures that are typically used for surface estimation, test information measures are more useful in experimental design for hypothesis testing and model selection. In particular, we obtain a probability based measure, which has more appealing properties than variance based measures in design contexts where decision problems are of interest. The underlying intuition of our design proposals is straightforward: to distinguish between models we should collect data from regions of the covariate space for which the models differ most. Nicolae et al. (2008) gave an asymptotic equivalence between their test information measures and Fisher information. We extend this result to all test information measures under our framework. Simulation studies and an application in astronomy demonstrate the utility of our approach, and provide comparison to other methods including that of Box and Hill (1967).

math.ST↗
arXiv2019

Warp Bridge Sampling: The Next Generation

Bridge sampling is an effective Monte Carlo method for estimating the ratio of normalizing constants of two probability densities, a routine computational problem in statistics, physics, chemistry, and other fields. The Monte Carlo error of the bridge sampling estimator is determined by the amount of overlap between the two densities. In the case of uni-modal densities, Warp-I, II, and III transformations (Meng and Schilling, 2002) are effective for increasing the initial overlap, but they are less so for multi-modal densities. This paper introduces Warp-U transformations that aim to transform multi-modal densities into Uni-modal ones without altering their normalizing constants. The construction of a Warp-U transformation starts with a Normal (or other convenient) mixture distribution $ϕ_{\text{mix}}$ that has reasonable overlap with the target density $p$, whose normalizing constant is unknown. The stochastic transformation that maps $ϕ_{\text{mix}}$ back to its generating distribution $N(0,1)$ is then applied to $p$ yielding its Warp-U version, which we denote $\tilde{p}$. Typically, $\tilde{p}$ is uni-modal and has substantially increased overlap with $N(0,1)$. Furthermore, we prove that the overlap between $\tilde{p}$ and $N(0,1)$ is guaranteed to be no less than the overlap between $p$ and $ϕ_{\text{mix}}$, in terms of any $f$-divergence. We propose a computationally efficient method to find an appropriate $ϕ_{\text{mix}}$, and a simple but effective approach to remove the bias which results from estimating the normalizing constants and fitting $ϕ_{\text{mix}}$ with the same data. We illustrate our findings using 10 and 50 dimensional highly irregular multi-modal densities, and demonstrate how Warp-U sampling can be used to improve the final estimation step of the Generalized Wang-Landau algorithm (Liang, 2005), a powerful sampling and estimation method.

stat.ME↗
arXiv2018

Calibration Concordance for Astronomical Instruments via Multiplicative Shrinkage

Calibration data are often obtained by observing several well-understood objects simultaneously with multiple instruments, such as satellites for measuring astronomical sources. Analyzing such data and obtaining proper concordance among the instruments is challenging when the physical source models are not well understood, when there are uncertainties in "known" physical quantities, or when data quality varies in ways that cannot be fully quantified. Furthermore, the number of model parameters increases with both the number of instruments and the number of sources. Thus, concordance of the instruments requires careful modeling of the mean signals, the intrinsic source differences, and measurement errors. In this paper, we propose a log-Normal hierarchical model and a more general log-t model that respect the multiplicative nature of the mean signals via a half-variance adjustment, yet permit imperfections in the mean modeling to be absorbed by residual variances. We present analytical solutions in the form of power shrinkage in special cases and develop reliable MCMC algorithms for general cases. We apply our method to several data sets obtained with a variety of X-ray telescopes such as Chandra. We demonstrate that our method provides helpful and practical guidance for astrophysicists when adjusting for disagreements among instruments.

stat.AP↗
arXiv2017

Judicious Judgment Meets Unsettling Updating: Dilation, Sure Loss, and Simpson's Paradox

Statistical learning using imprecise probabilities is gaining more attention because it presents an alternative strategy for reducing irreplicable findings by freeing the user from the task of making up unwarranted high-resolution assumptions. However, model updating as a mathematical operation is inherently exact, hence updating imprecise models requires the user's judgment in choosing among competing updating rules. These rules often lead to incompatible inferences, and can exhibit unsettling phenomena like dilation, contraction and sure loss, which cannot occur with the Bayes rule and precise probabilities. We revisit a number of famous "paradoxes", including the three prisoners/Monty Hall problem, revealing that a logical fallacy arises from a set of marginally plausible yet jointly incommensurable assumptions when updating the underlying imprecise model. We establish an equivalence between Simpson's paradox and an implicit adoption of a pair of aggregation rules that induce sure loss. We also explore behavioral discrepancies between the generalized Bayes rule, Dempster's rule and the Geometric rule as alternative posterior updating rules for Choquet capacities of order 2. We show that both the generalized Bayes rule and Geometric rule are incapable of updating without prior information regardless of how strong the information in our data is, and that Dempster's rule and the Geometric rule can mathematically contradict each other with respect to dilation and contraction. Our findings show that unsettling updates reflect a collision between the rules' assumptions and the inexactness allowed by the model itself, highlighting the invaluable role of judicious judgment in handling low-resolution information, and the care we must take when applying learning rules to update imprecise probabilities.

math.ST↗
arXiv2017

Conducting Highly Principled Data Science: A Statistician's Job and Joy

Highly Principled Data Science insists on methodologies that are: (1) scientifically justified, (2) statistically principled, and (3) computationally efficient. An astrostatistics collaboration, together with some reminiscences, illustrates the increased roles statisticians can and should play to ensure this trio, and to advance the science of data along the way.

stat.OT↗
arXiv2017

A Repelling-Attracting Metropolis Algorithm for Multimodality

Although the Metropolis algorithm is simple to implement, it often has difficulties exploring multimodal distributions. We propose the repelling-attracting Metropolis (RAM) algorithm that maintains the simple-to-implement nature of the Metropolis algorithm, but is more likely to jump between modes. The RAM algorithm is a Metropolis-Hastings algorithm with a proposal that consists of a downhill move in density that aims to make local modes repelling, followed by an uphill move in density that aims to make local modes attracting. The downhill move is achieved via a reciprocal Metropolis ratio so that the algorithm prefers downward movement. The uphill move does the opposite using the standard Metropolis ratio which prefers upward movement. This down-up movement in density increases the probability of a proposed move to a different mode. Because the acceptance probability of the proposal involves a ratio of intractable integrals, we introduce an auxiliary variable which creates a term in the acceptance probability that cancels with the intractable ratio. Using several examples, we demonstrate the potential for the RAM algorithm to explore a multimodal distribution more efficiently than a Metropolis algorithm and with less tuning than is commonly required by tempering-based methods.

stat.ME↗
arXiv2017

Bayesian Estimates of Astronomical Time Delays between Gravitationally Lensed Stochastic Light Curves

The gravitational field of a galaxy can act as a lens and deflect the light emitted by a more distant object such as a quasar. Strong gravitational lensing causes multiple images of the same quasar to appear in the sky. Since the light in each gravitationally lensed image traverses a different path length from the quasar to the Earth, fluctuations in the source brightness are observed in the several images at different times. The time delay between these fluctuations can be used to constrain cosmological parameters and can be inferred from the time series of brightness data or light curves of each image. To estimate the time delay, we construct a model based on a state-space representation for irregularly observed time series generated by a latent continuous-time Ornstein-Uhlenbeck process. We account for microlensing, an additional source of independent long-term extrinsic variability, via a polynomial regression. Our Bayesian strategy adopts a Metropolis-Hastings within Gibbs sampler. We improve the sampler by using an ancillarity-sufficiency interweaving strategy and adaptive Markov chain Monte Carlo. We introduce a profile likelihood of the time delay as an approximation of its marginal posterior distribution. The Bayesian and profile likelihood approaches complement each other, producing almost identical results; the Bayesian method is more principled but the profile likelihood is simpler to implement. We demonstrate our estimation strategy using simulated data of doubly- and quadruply-lensed quasars, and observed data from quasars Q0957+561 and J1029+2623.

astro-ph.IM↗
arXiv2015

There is Individualized Treatment. Why Not Individualized Inference?

Doctors use statistics to advance medical knowledge; we use a medical analogy to introduce statistical inference "from scratch" and to highlight an improvement. Your doctor, perhaps implicitly, predicts the effectiveness of a treatment for you based on its performance in a clinical trial; the trial patients serve as controls for you. The same logic underpins statistical inference: to identify the best statistical procedure to use for a problem, we simulate a set of control problems and evaluate candidate procedures on the controls. Now for the improvement: recent interest in personalized/individualized medicine stems from the recognition that some clinical trial patients are better controls for you than others. Therefore, treatment decisions for you should depend only on a subset of relevant patients. Individualized statistical inference implements this idea for control problems (rather than patients). Its potential for improving data analysis matches personalized medicine's for improving healthcare. The central issue--for both individualized medicine and individualized inference--is how to make the right relevance robustness trade-off: if we exercise too much judgement in determining which controls are relevant, our inferences will not be robust. How much is too much? We argue that the unknown answer is the Holy Grail of statistical inference.

stat.ME↗
arXiv2015

An unexpected encounter with Cauchy and Lévy

The Cauchy distribution is usually presented as a mathematical curiosity, an exception to the Law of Large Numbers, or even as an "Evil" distribution in some introductory courses. It therefore surprised us when Drton and Xiao (2016) proved the following result for $m=2$ and conjectured it for $m\ge 3$. Let $X= (X_1,..., X_m)$ and $Y = (Y_1, ...,Y_m)$ be i.i.d $N(0,Σ)$, where $Σ=\{σ_{ij}\}\ge 0$ is an $m\times m$ and \textit{arbitrary} covariance matrix with $σ_{jj}>0$ for all $1\leq j\leq m$. Then $$Z = \sum_{j=1}^m w_j \frac{X_j}{Y_j} \ \sim \mathrm{Cauchy}(0,1),$$ as long as $w=(w_1,..., w_m) $ is independent of $(X, Y)$, $w_j\ge 0, j=1,..., m$, and $\sum_{j=1}^m w_j=1$. In this note, we present an elementary proof of this conjecture for any $m \geq 2$ by linking $Z$ to a geometric characterization of Cauchy(0,1) given in Willams (1969). This general result is essential to the large sample behavior of Wald tests in many applications such as factor models and contingency tables. It also leads to other unexpected results such as $$ \sum_{i=1}^m\sum_{j=1}^m \frac{w_iw_jσ_{ij}}{X_iX_j} \sim {\text{Lévy}}(0, 1). $$ This generalizes the "super Cauchy phenomenon" that the average of $m$ i.i.d. standard Lévy variables (i.e., inverse chi-squared variables with one degree of freedom) has the same distribution as that of a single standard Lévy variable multiplied by $m$ (which is obtained by taking $w_j=1/m$ and $Σ$ to be the identity matrix).

math.ST↗
arXiv2014

Strong Lens Time Delay Challenge: II. Results of TDC1

We present the results of the first strong lens time delay challenge. The motivation, experimental design, and entry level challenge are described in a companion paper. This paper presents the main challenge, TDC1, which consisted of analyzing thousands of simulated light curves blindly. The observational properties of the light curves cover the range in quality obtained for current targeted efforts (e.g.,~COSMOGRAIL) and expected from future synoptic surveys (e.g.,~LSST), and include simulated systematic errors. \nteamsA\ teams participated in TDC1, submitting results from \nmethods\ different method variants. After a describing each method, we compute and analyze basic statistics measuring accuracy (or bias) $A$, goodness of fit $χ^2$, precision $P$, and success rate $f$. For some methods we identify outliers as an important issue. Other methods show that outliers can be controlled via visual inspection or conservative quality control. Several methods are competitive, i.e., give $|A|<0.03$, $P<0.03$, and $χ^2<1.5$, with some of the methods already reaching sub-percent accuracy. The fraction of light curves yielding a time delay measurement is typically in the range $f = $20--40\%. It depends strongly on the quality of the data: COSMOGRAIL-quality cadence and light curve lengths yield significantly higher $f$ than does sparser sampling. Taking the results of TDC1 at face value, we estimate that LSST should provide around 400 robust time-delay measurements, each with $P<0.03$ and $|A|<0.01$, comparable to current lens modeling uncertainties. In terms of observing strategies, we find that $A$ and $f$ depend mostly on season length, while P depends mostly on cadence and campaign duration.

astro-ph.IM↗
arXiv2013

The potential and perils of preprocessing: Building new foundations

Preprocessing forms an oft-neglected foundation for a wide range of statistical and scientific analyses. However, it is rife with subtleties and pitfalls. Decisions made in preprocessing constrain all later analyses and are typically irreversible. Hence, data analysis becomes a collaborative endeavor by all parties involved in data collection, preprocessing and curation, and downstream inference. Even if each party has done its best given the information and resources available to them, the final result may still fall short of the best possible in the traditional single-phase inference framework. This is particularly relevant as we enter the era of "big data". The technologies driving this data explosion are subject to complex new forms of measurement error. Simultaneously, we are accumulating increasingly massive databases of scientific analyses. As a result, preprocessing has become more vital (and potentially more dangerous) than ever before.

math.ST↗
arXiv2011

Cross-Fertilizing Strategies for Better EM Mountain Climbing and DA Field Exploration: A Graphical Guide Book

In recent years, a variety of extensions and refinements have been developed for data augmentation based model fitting routines. These developments aim to extend the application, improve the speed and/or simplify the implementation of data augmentation methods, such as the deterministic EM algorithm for mode finding and stochastic Gibbs sampler and other auxiliary-variable based methods for posterior sampling. In this overview article we graphically illustrate and compare a number of these extensions, all of which aim to maintain the simplicity and computation stability of their predecessors. We particularly emphasize the usefulness of identifying similarities between the deterministic and stochastic counterparts as we seek more efficient computational strategies. We also demonstrate the applicability of data augmentation methods for handling complex models with highly hierarchical structure, using a high-energy high-resolution spectral imaging model for data from satellite telescopes, such as the Chandra X-ray Observatory.

stat.ME↗
arXiv2011

Accounting for Calibration Uncertainties in X-ray Analysis: Effective Areas in Spectral Fitting

While considerable advance has been made to account for statistical uncertainties in astronomical analyses, systematic instrumental uncertainties have been generally ignored. This can be crucial to a proper interpretation of analysis results because instrumental calibration uncertainty is a form of systematic uncertainty. Ignoring it can underestimate error bars and introduce bias into the fitted values of model parameters. Accounting for such uncertainties currently requires extensive case-specific simulations if using existing analysis packages. Here we present general statistical methods that incorporate calibration uncertainties into spectral analysis of high-energy data. We first present a method based on multiple imputation that can be applied with any fitting method, but is necessarily approximate. We then describe a more exact Bayesian approach that works in conjunction with a Markov chain Monte Carlo based fitting. We explore methods for improving computational efficiency, and in particular detail a method of summarizing calibration uncertainties with a principal component analysis of samples of plausible calibration files. This method is implemented using recently codified Chandra effective area uncertainties for low-resolution spectral analysis and is verified using both simulated and actual Chandra data. Our procedure for incorporating effective area uncertainty is easily generalized to other types of calibration uncertainties.

astro-ph.IM↗
arXiv2011

Quantifying the Fraction of Missing Information for Hypothesis Testing in Statistical and Genetic Studies

Many practical studies rely on hypothesis testing procedures applied to data sets with missing information. An important part of the analysis is to determine the impact of the missing data on the performance of the test, and this can be done by properly quantifying the relative (to complete data) amount of available information. The problem is directly motivated by applications to studies, such as linkage analyses and haplotype-based association projects, designed to identify genetic contributions to complex diseases. In the genetic studies the relative information measures are needed for the experimental design, technology comparison, interpretation of the data, and for understanding the behavior of some of the inference tools. The central difficulties in constructing such information measures arise from the multiple, and sometimes conflicting, aims in practice. For large samples, we show that a satisfactory, likelihood-based general solution exists by using appropriate forms of the relative Kullback--Leibler information, and that the proposed measures are computationally inexpensive given the maximized likelihoods with the observed data. Two measures are introduced, under the null and alternative hypothesis respectively. We exemplify the measures on data coming from mapping studies on the inflammatory bowel disease and diabetes. For small-sample problems, which appear rather frequently in practice and sometimes in disguised forms (e.g., measuring individual contributions to a large study), the robust Bayesian approach holds great promise, though the choice of a general-purpose "default prior" is a very challenging problem.

stat.ME↗
arXiv2010

Decoding the H-likelihood

Discussion of "Likelihood Inference for Models with Unobservables: Another View" by Youngjo Lee and John A. Nelder [arXiv:1010.0303]

stat.ME↗