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Jan Hannig

Publications and source records attributed to Jan Hannig.

At least 37 records · Page 2Linked to original sources
arXiv2023

A Geometric Perspective on Bayesian and Generalized Fiducial Inference

Post-data statistical inference concerns making probability statements about model parameters conditional on observed data. When a priori knowledge about parameters is available, post-data inference can be conveniently made from Bayesian posteriors. In the absence of prior information, we may still rely on objective Bayes or generalized fiducial inference (GFI). Inspired by approximate Bayesian computation, we propose a novel characterization of post-data inference with the aid of differential geometry. Under suitable smoothness conditions, we establish that Bayesian posteriors and generalized fiducial distributions (GFDs) can be respectively characterized by absolutely continuous distributions supported on the same differentiable manifold: The manifold is uniquely determined by the observed data and the data generating equation of the fitted model. Our geometric analysis not only sheds light on the connection and distinction between Bayesian inference and GFI, but also allows us to sample from posteriors and GFDs using manifold Markov chain Monte Carlo algorithms. A repeated-measures analysis of variance example is presented to illustrate the sampling procedure.

math.ST↗
arXiv2023

Measure of Strength of Evidence for Visually Observed Differences between Subpopulations

For measuring the strength of visually-observed subpopulation differences, the Population Difference Criterion is proposed to assess the statistical significance of visually observed subpopulation differences. It addresses the following challenges: in high-dimensional contexts, distributional models can be dubious; in high-signal contexts, conventional permutation tests give poor pairwise comparisons. We also make two other contributions: Based on a careful analysis we find that a balanced permutation approach is more powerful in high-signal contexts than conventional permutations. Another contribution is the quantification of uncertainty due to permutation variation via a bootstrap confidence interval. The practical usefulness of these ideas is illustrated in the comparison of subpopulations of modern cancer data.

stat.ME↗
arXiv2023

Introduction to Generalized Fiducial Inference

Fiducial inference was introduced in the first half of the 20th century by Fisher (1935) as a means to get a posterior-like distribution for a parameter without having to arbitrarily define a prior. While the method originally fell out of favor due to non-exactness issues in multivariate cases, the method has garnered renewed interest in the last decade. This is partly due to the development of generalized fiducial inference, which is a fiducial perspective on generalized confidence intervals: a method used to find approximate confidence distributions. In this chapter, we illuminate the usefulness of the fiducial philosophy, introduce the definition of a generalized fiducial distribution, and apply it to interesting, non-trivial inferential examples.

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arXiv2022

A fiducial approach to nonparametric deconvolution problem: discrete case

Fiducial inference, as generalized by Hannig et al. (2016), is applied to nonparametric g-modeling (Efron, 2016) in the discrete case. We propose a computationally efficient algorithm to sample from the fiducial distribution, and use the generated samples to construct point estimates and confidence intervals. We study the theoretical properties of the fiducial distribution and perform extensive simulations in various scenarios. The proposed approach yields good statistical performance in terms of the mean squared error of point estimators and the coverage of confidence intervals. Furthermore, we apply the proposed fiducial method to estimate the probability of each satellite site being malignant using gastric adenocarcinoma data with 844 patients (Efron, 2016).

stat.ME↗
arXiv2022

Generalized Fiducial Inference on Differentiable Manifolds

We introduce a novel approach to inference on parameters that take values in a Riemannian manifold embedded in a Euclidean space. Parameter spaces of this form are ubiquitous across many fields, including chemistry, physics, computer graphics, and geology. This new approach uses generalized fiducial inference to obtain a posterior-like distribution on the manifold, without needing to know a parameterization that maps the constrained space to an unconstrained Euclidean space. The proposed methodology, called the constrained generalized fiducial distribution (CGFD), is obtained by using mathematical tools from Riemannian geometry. A Bernstein-von Mises-type result for the CGFD, which provides intuition for how the desirable asymptotic qualities of the unconstrained generalized fiducial distribution are inherited by the CGFD, is provided. To demonstrate the practical use of the CGFD, we provide three proof-of-concept examples: inference for data from a multivariate normal density with the mean parameters on a sphere, a linear logspline density estimation problem, and a reimagined approach to the AR(1) model, all of which exhibit desirable coverages via simulation. We discuss two Markov chain Monte Carlo algorithms for the exploration of these constrained parameter spaces and adapt them for the CGFD.

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arXiv2022

Jackstraw Inference for AJIVE Data Integration

In the age of big data, data integration is a critical step especially in the understanding of how diverse data types work together and work separately. Among data integration methods, the Angle-Based Joint and Individual Variation Explained (AJIVE) approach is particularly attractive because it not only studies joint behavior but also individual behavior. Typically AJIVE scores indicate important relationships between data objects, such as clusters. An important challenge is understanding which features, i.e. variables, are associated with those relationships. This challenge is addressed by the proposal of a hypothesis test for assessing statistical significance of features. The new test is inspired by the related jackstraw method developed for Principal Component Analysis. We use a high-dimensional muti-genomic cancer data set as our strong motivation and deep illustration of the methodology.

stat.AP↗
arXiv2022

Demystifying Inferential Models: A Fiducial Perspective

Inferential models have recently gained in popularity for valid uncertainty quantification. In this paper, we investigate inferential models by exploring relationships between inferential models, fiducial inference, and confidence curves. In short, we argue that from a certain point of view, inferential models can be viewed as fiducial distribution based confidence curves. We show that all probabilistic uncertainty quantification of inferential models is based on a collection of sets we name principle sets and principle assertions.

stat.ME↗
arXiv2022

A New String Edit Distance and Applications

String edit distances have been used for decades in applications ranging from spelling correction and web search suggestions to DNA analysis. Most string edit distances are variations of the Levenshtein distance and consider only single-character edits. In forensic applications polymorphic genetic markers such as short tandem repeats (STRs) are used. At these repetitive motifs the DNA copying errors consist of more than just single base differences. More often the phenomenon of ``stutter'' is observed, where the number of repeated units differs (by whole units) from the template. To adapt the Levenshtein distance to be suitable for forensic applications where DNA sequence similarity is of interest, a generalized string edit distance is defined that accommodates the addition or deletion of whole motifs in addition to single-nucleotide edits. A dynamic programming implementation is developed for computing this distance between sequences. The novelty of this algorithm is in handling the complex interactions that arise between multiple- and single-character edits. Forensic examples illustrate the purpose and use of the Restricted Forensic Levenshtein (RFL) distance measure, but applications extend to sequence alignment and string similarity in other biological areas, as well as dynamic programming algorithms more broadly.

q-bio.GN↗
arXiv2021

A unified nonparametric fiducial approach to interval-censored data

Censored data, where the event time is partially observed, are challenging for survival probability estimation. In this paper, we introduce a novel nonparametric fiducial approach to interval-censored data, including right-censored, current status, case II censored, and mixed case censored data. The proposed approach leveraging a simple Gibbs sampler has a useful property of being "one size fits all", i.e., the proposed approach automatically adapts to all types of non-informative censoring mechanisms. As shown in the extensive simulations, the proposed fiducial confidence intervals significantly outperform existing methods in terms of both coverage and length. In addition, the proposed fiducial point estimator has much smaller estimation errors than the nonparametric maximum likelihood estimator. Furthermore, we apply the proposed method to Austrian rubella data and a study of hemophiliacs infected with the human immunodeficiency virus. The strength of the proposed fiducial approach is not only estimation and uncertainty quantification but also its automatic adaptation to a variety of censoring mechanisms.

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arXiv2021

New Perspectives on Centering

Data matrix centering is an ever-present yet under-examined aspect of data analysis. Functional data analysis (FDA) often operates with a default of centering such that the vectors in one dimension have mean zero. We find that centering along the other dimension identifies a novel useful mode of variation beyond those familiar in FDA. We explore ambiguities in both matrix orientation and nomenclature. Differences between centerings and their potential interaction can be easily misunderstood. We propose a unified framework and new terminology for centering operations. We clearly demonstrate the intuition behind and consequences of each centering choice with informative graphics. We also propose a new direction energy hypothesis test as part of a series of diagnostics for determining which choice of centering is best for a data set. We explore the application of these diagnostics in several FDA settings.

stat.ME↗
arXiv2020

Generalized fiducial factor: an alternative to the Bayes factor for forensic identification of source problems

One formulation of forensic identification of source problems is to determine the source of trace evidence, for instance, glass fragments found on a suspect for a crime. The current state of the science is to compute a Bayes factor (BF) comparing the marginal distribution of measurements of trace evidence under two competing propositions for whether or not the unknown source evidence originated from a specific source. The obvious problem with such an approach is the ability to tailor the prior distributions (placed on the features/parameters of the statistical model for the measurements of trace evidence) in favor of the defense or prosecution, which is further complicated by the fact that the typical number of measurements of trace evidence is typically sufficiently small that prior choice/specification has a strong influence on the value of the BF. To remedy this problem of prior specification and choice, we develop an alternative to the BF, within the framework of generalized fiducial inference (GFI), that we term a {\em generalized fiducial factor} (GFF). Furthermore, we demonstrate empirically, on the synthetic and real Netherlands Forensic Institute (NFI) casework data, deficiencies in the BF and classical/frequentist likelihood ratio (LR) approaches.

stat.ME↗
arXiv2020

Deep Fiducial Inference

Since the mid-2000s, there has been a resurrection of interest in modern modifications of fiducial inference. To date, the main computational tool to extract a generalized fiducial distribution is Markov chain Monte Carlo (MCMC). We propose an alternative way of computing a generalized fiducial distribution that could be used in complex situations. In particular, to overcome the difficulty when the unnormalized fiducial density (needed for MCMC), we design a fiducial autoencoder (FAE). The fitted autoencoder is used to generate generalized fiducial samples of the unknown parameters. To increase accuracy, we then apply an approximate fiducial computation (AFC) algorithm, by rejecting samples that when plugged into a decoder do not replicate the observed data well enough. Our numerical experiments show the effectiveness of our FAE-based inverse solution and the excellent coverage performance of the AFC corrected FAE solution.

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arXiv2020

Subspace Clustering through Sub-Clusters

The problem of dimension reduction is of increasing importance in modern data analysis. In this paper, we consider modeling the collection of points in a high dimensional space as a union of low dimensional subspaces. In particular we propose a highly scalable sampling based algorithm that clusters the entire data via first spectral clustering of a small random sample followed by classifying or labeling the remaining out of sample points. The key idea is that this random subset borrows information across the entire data set and that the problem of clustering points can be replaced with the more efficient and robust problem of "clustering sub-clusters". We provide theoretical guarantees for our procedure. The numerical results indicate we outperform other state-of-the-art subspace clustering algorithms with respect to accuracy and speed.

stat.ML↗
arXiv2020

Joint and individual analysis of breast cancer histologic images and genomic covariates

A key challenge in modern data analysis is understanding connections between complex and differing modalities of data. For example, two of the main approaches to the study of breast cancer are histopathology (analyzing visual characteristics of tumors) and genetics. While histopathology is the gold standard for diagnostics and there have been many recent breakthroughs in genetics, there is little overlap between these two fields. We aim to bridge this gap by developing methods based on Angle-based Joint and Individual Variation Explained (AJIVE) to directly explore similarities and differences between these two modalities. Our approach exploits Convolutional Neural Networks (CNNs) as a powerful, automatic method for image feature extraction to address some of the challenges presented by statistical analysis of histopathology image data. CNNs raise issues of interpretability that we address by developing novel methods to explore visual modes of variation captured by statistical algorithms (e.g. PCA or AJIVE) applied to CNN features. Our results provide many interpretable connections and contrasts between histopathology and genetics.

q-bio.QM↗
arXiv2019

Uncertainty Quantification in Ensembles of Honest Regression Trees using Generalized Fiducial Inference

Due to their accuracies, methods based on ensembles of regression trees are a popular approach for making predictions. Some common examples include Bayesian additive regression trees, boosting and random forests. This paper focuses on honest random forests, which add honesty to the original form of random forests and are proved to have better statistical properties. The main contribution is a new method that quantifies the uncertainties of the estimates and predictions produced by honest random forests. The proposed method is based on the generalized fiducial methodology, and provides a fiducial density function that measures how likely each single honest tree is the true model. With such a density function, estimates and predictions, as well as their confidence/prediction intervals, can be obtained. The promising empirical properties of the proposed method are demonstrated by numerical comparisons with several state-of-the-art methods, and by applications to a few real data sets. Lastly, the proposed method is theoretically backed up by a strong asymptotic guarantee.

stat.ME↗
arXiv2019

A Note on Optimal Sampling Strategy for Structural Variant Detection Using Optical Mapping

Structural variants compose the majority of human genetic variation, but are difficult to assess using current genomic sequencing technologies. Optical mapping technologies, which measure the size of chromosomal fragments between labeled markers, offer an alternative approach. As these technologies mature towards becoming clinical tools, there is a need to develop an approach for determining the optimal strategy for sampling biological material in order to detect a variant at some threshold. Here we develop an optimization approach using a simple, yet realistic, model of the genomic mapping process using a hyper-geometric distribution and {probabilistic} concentration inequalities. Our approach is both computationally and analytically tractable and includes a novel approach to getting tail bounds of hyper-geometric distribution. We show that if a genomic mapping technology can sample most of the chromosomal fragments within a sample, comparatively little biological material is needed to detect a variant at high confidence.

q-bio.QM↗
arXiv2019

The EAS approach for graphical selection consistency in vector autoregression models

As evidenced by various recent and significant papers within the frequentist literature, along with numerous applications in macroeconomics, genomics, and neuroscience, there continues to be substantial interest to understand the theoretical estimation properties of high-dimensional vector autoregression (VAR) models. To date, however, while Bayesian VAR (BVAR) models have been developed and studied empirically (primarily in the econometrics literature) there exist very few theoretical investigations of the repeated sampling properties for BVAR models in the literature. In this direction, we construct methodology via the $\varepsilon$-$admissible$ subsets (EAS) approach for posterior-like inference based on a generalized fiducial distribution of relative model probabilities over all sets of active/inactive components (graphs) of the VAR transition matrix. We provide a mathematical proof of $pairwise$ and $strong$ graphical selection consistency for the EAS approach for stable VAR(1) models which is robust to model misspecification, and demonstrate numerically that it is an effective strategy in high-dimensional settings.

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