We consider Bayesian nonparametric inference in the right-censoring survival model, where modeling is made at the level of the hazard rate. We derive posterior limiting distributions for linear functionals of the hazard, and then for `many' functionals simultaneously in appropriate multiscale spaces. As an application, we derive Bernstein-von Mises theorems for the cumulative hazard and survival functions, which lead to asymptotically efficient confidence bands for these quantities. Further, we show optimal posterior contraction rates for the hazard in terms of the supremum norm. In medical studies, a popular approach is to model hazards a priori as random histograms with possibly dependent heights. This and more general classes of arbitrarily smooth prior distributions are considered as applications of our theory. A sampler is provided for possibly dependent histogram posteriors. Its finite sample properties are investigated on both simulated and real data experiments.
A new computation method of frequentist $p$-values and Bayesian posterior probabilities based on the bootstrap probability is discussed for the multivariate normal model with unknown expectation parameter vector. The null hypothesis is represented as an arbitrary-shaped region. We introduce new parametric models for the scaling-law of bootstrap probability so that the multiscale bootstrap method, which was designed for one-sided test, can also computes confidence measures of two-sided test, extending applicability to a wider class of hypotheses. Parameter estimation is improved by the two-step…
This paper takes a quick look at Bayesian joint models (BJM) for longitudinal and survival data. A general formulation for BJM is examined in terms of the sampling distribution of the longitudinal and survival processes, the conditional distribution of the random effects and the prior distribution. Next a basic BJM defined in terms of a mixed linear model and a Cox survival regression models is discussed and some extensions and other Bayesian topics are briefly outlined.
An Archimedean copula is characterised by its generator. This is a real function whose inverse behaves as a survival function. We propose a semiparametric generator based on a quadratic spline. This is achieved by modelling the first derivative of a hazard rate function, in a survival analysis context, as a piecewise constant function. Convexity of our semiparametric generator is obtained by imposing some simple constraints. The induced semiparametric Archimedean copula produces Kendall's tau association measure that covers the whole range $(-1,1)$. Inference on the model is done under a Bayes…
We tackle the problem of multiscale regression for predictors that are spatially or temporally indexed, or with a pre-specified multiscale structure, with a Bayesian modular approach. The regression function at the finest scale is expressed as an additive expansion of coarse to fine step functions. Our Modular and Multiscale (M&M) methodology provides multiscale decomposition of high-dimensional data arising from very fine measurements. Unlike more complex methods for functional predictors, our approach provides easy interpretation of the results. Additionally, it provides a quantification of …
We propose a Multiscale Invertible Generative Network (MsIGN) and associated training algorithm that leverages multiscale structure to solve high-dimensional Bayesian inference. To address the curse of dimensionality, MsIGN exploits the low-dimensional nature of the posterior, and generates samples from coarse to fine scale (low to high dimension) by iteratively upsampling and refining samples. MsIGN is trained in a multi-stage manner to minimize the Jeffreys divergence, which avoids mode dropping in high-dimensional cases. On two high-dimensional Bayesian inverse problems, we show superior pe…
In many inverse problems, model parameters cannot be precisely determined from observational data. Bayesian inference provides a mechanism for capturing the resulting parameter uncertainty, but typically at a high computational cost. This work introduces a multiscale decomposition that exploits conditional independence across scales, when present in certain classes of inverse problems, to decouple Bayesian inference into two stages: (1) a computationally tractable coarse-scale inference problem; and (2) a mapping of the low-dimensional coarse-scale posterior distribution into the original high…
Recently, the Bayesian nonparametric approach in survival studies attracts much more attentions. Because of multi modality in survival data, the mixture models are very common in this field. One of the famous priors on Bayesian nonparametric models is Dirichlet process prior. In this paper we introduce a Bayesian nonparametric mixture model with Burr distribution(Burr type XII) as the kernel of mixture model. Since the Burr distribution shares good properties of common distributions on survival analysis, it has more flexibility than other distributions. By applying this model to simulated and …
Robust Survival Estimation under Interval Censoring: Expectation-Maximization and Bayesian Accelerated Failure Time Assessment via Simulation and Application
Popular parametric and semiparametric hazards regression models for clustered survival data are inappropriate and inadequate when the unknown effects of different covariates and clustering are complex. This calls for a flexible modeling framework to yield efficient survival prediction. Moreover, for some survival studies involving time to occurrence of some asymptomatic events, survival times are typically interval censored between consecutive clinical inspections. In this article, we propose a robust semiparametric model for clustered interval-censored survival data under a paradigm of Bayesi…
In many real problems, dependence structures more general than exchangeability are required. For instance, in some settings partial exchangeability is a more reasonable assumption. For this reason, vectors of dependent Bayesian nonparametric priors have recently gained popularity. They provide flexible models which are tractable from a computational and theoretical point of view. In this paper, we focus on their use for estimating survival functions with multiple-samples information. Our methodology allows to model the dependence among survival times of different groups of observations and ext…
Objective: To provide guidance for the use of the main functions available in R for performing post hoc Bayesian analysis of a randomized clinical trial with a survival endpoint using proportional hazard models. Study Design and Setting: Data derived from the ALLOZITHRO trial, conducted with 465 patients after allograft to prevent pulmonary complications and allocated between azithromycin and placebo; airflow decline-free survival at 2 years after randomization was the main endpoint. Results: Despite heterogeneity in modeling assumptions, in particular for the baseline hazard (parametric or no…
In this work, we propose a new Bayesian spatial homogeneity pursuit method for survival data under the proportional hazards model to detect spatially clustered patterns in baseline hazard and regression coefficients. Specially, regression coefficients and baseline hazard are assumed to have spatial homogeneity pattern over space. To capture such homogeneity, we develop a geographically weighted Chinese restaurant process prior to simultaneously estimate coefficients and baseline hazards and their uncertainty measures. An efficient Markov chain Monte Carlo (MCMC) algorithm is designed for our p…
The aim of this paper is to model the length of registration at university and its associated academic outcome for undergraduate students at the Pontificia Universidad Católica de Chile. Survival time is defined as the time until the end of the enrollment period, which can relate to different reasons - graduation or two types of dropout - that are driven by different processes. Hence, a competing risks model is employed for the analysis. The issue of separation of the outcomes (which precludes maximum likelihood estimation) is handled through the use of Bayesian inference with an appropriately…
We study objective Bayesian inference for linear regression models with residual errors distributed according to the class of two-piece scale mixtures of normal distributions. These models allow for capturing departures from the usual assumption of normality of the errors in terms of heavy tails, asymmetry, and certain types of heteroscedasticity. We propose a general noninformative, scale-invariant, prior structure and provide sufficient conditions for the propriety of the posterior distribution of the model parameters, which cover cases when the response variables are censored. These results…
In Bayesian semi-parametric analyses of time-to-event data, non-parametric process priors are adopted for the baseline hazard function or the cumulative baseline hazard function for a given finite partition of the time axis. However, it would be controversial to suggest a general guideline to construct an optimal time partition. While a great deal of research has been done to relax the assumption of the fixed split times for other non-parametric processes, to our knowledge, no methods have been developed for a gamma process prior, which is one of the most widely used in Bayesian survival analy…
We develop a flexible Erlang mixture model for survival analysis. The model for the survival density is built from a structured mixture of Erlang densities, mixing on the integer shape parameter with a common scale parameter. The mixture weights are constructed through increments of a distribution function on the positive real line, which is assigned a Dirichlet process prior. The model has a relatively simple structure, balancing flexibility with efficient posterior computation. Moreover, it implies a mixture representation for the hazard function that involves time-dependent mixture weights,…
A Generalized Framework for Multiscale State-Space Modeling with Nested Nonlinear Dynamics: An Application to Bayesian Learning under Switching Regimes
School of Complex Adaptive Systems（复杂适应系统学院）；Arizona State University（亚利桑那州立大学）
We built a novel Bayesian hierarchical survival model based on the somatic mutation profile of patients across 50 genes and 27 cancer types. The pan-cancer quality allows for the model to "borrow" information across cancer types, motivated by the assumption that similar mutation profiles may have similar (but not necessarily identical) effects on survival across different tissues-of-origin or tumor types. The effect of a mutation at each gene was allowed to vary by cancer type while the mean effect of each gene was shared across cancers. Within this framework we considered four parametric surv…
To better understand effects of exposure to food allergens, food challenge studies are designed to slowly increase the dose of an allergen delivered to allergic individuals until an objective reaction occurs. These dose-to-failure studies are used to determine acceptable intake levels and are analyzed using parametric failure time models. Though these models can provide estimates of the survival curve, their parametric form may misrepresent the survival function for doses of interest, and different models that describe the data similarly may produce different dose-to-failure estimates. Motivat…
Bayesian paradigm takes advantage of well fitting complicated survival models and feasible computing in survival analysis owing to the superiority in tackling the complex censoring scheme, compared with the frequentist paradigm. In this chapter, we aim to display the latest tendency in Bayesian computing, in the sense of automating the posterior sampling, through Bayesian analysis of survival modeling for multivariate survival outcomes with complicated data structure. Motivated by relaxing the strong assumption of proportionality and the restriction of a common baseline population, we propose …
Survival models are used to analyze time-to-event data in a variety of disciplines. Proportional hazard models provide interpretable parameter estimates, but proportional hazards assumptions are not always appropriate. Non-parametric models are more flexible but often lack a clear inferential framework. We propose a Bayesian tree partition model which is both flexible and inferential. Inference is obtained through the posterior tree structure and flexibility is preserved by modeling the the hazard function in each partition using a latent exponentiated Gaussian process. An efficient reversible…
Bayesian nonparametric marginal methods are very popular since they lead to fairly easy implementation due to the formal marginalization of the infinite-dimensional parameter of the model. However, the straightforwardness of these methods also entails some limitations: they typically yield point estimates in the form of posterior expectations, but cannot be used to estimate non-linear functionals of the posterior distribution, such as median, mode or credible intervals. This is particularly relevant in survival analysis where non-linear functionals such as e.g. the median survival time, play a…
Quantifying uncertainty of individualized treatment effects in right-censored survival data: A comparison of Bayesian additive regression trees and causal survival forest
While existing mathematical descriptions can accurately account for phenomena at microscopic scales (e.g. molecular dynamics), these are often high-dimensional, stochastic and their applicability over macroscopic time scales of physical interest is computationally infeasible or impractical. In complex systems, with limited physical insight on the coherent behavior of their constituents, the only available information is data obtained from simulations of the trajectories of huge numbers of degrees of freedom over microscopic time scales. This paper discusses a Bayesian approach to deriving prob…
In this paper, we propose a non-parametric Bayesian network method to generate synthetic scenarios of hourly generation for variable renewable energy(VRE) plants. The methodology consists of a non-parametric estimation of the probability distribution of VRE generation, followed by an inverse probability integral transform, in order to obtain normally distributed variables of VRE generation. Then, we build a Bayesian network based on the evaluation of the spatial correlation between variables (VRE generation and hydro inflows, but load forecast, temperature, and other types of random variables …
Bayesian nonparametric inferential procedures based on Markov chain Monte Carlo marginal methods typically yield point estimates in the form of posterior expectations. Though very useful and easy to implement in a variety of statistical problems, these methods may suffer from some limitations if used to estimate non-linear functionals of the posterior distribution. The main goal of the present paper is to develop a novel methodology that extends a well-established marginal procedure designed for hazard mixture models, in order to draw approximate inference on survival functions that is not lim…
Determining the extent to which a patient is benefiting from cancer therapy is challenging. Criteria for quantifying the extent of "tumor response" observed within a few cycles of treatment have been established for various types of solid as well as hematologic malignancies. These measures comprise the primary endpoints of phase II trials. Regulatory approvals of new cancer therapies, however, are usually contingent upon the demonstration of superior overall survival with randomized evidence acquired with a phase III trial comparing the novel therapy to an appropriate standard of care treatmen…
An evaluation of FBST, Fully Bayesian Significance Test, restricted to survival models is the main objective of the present paper. A Survival distribution should be chosen among the tree celebrated ones, lognormal, gamma, and Weibull. For this discrimination, a linear mixture of the three distributions, for which the mixture weights are defined by a Dirichlet distribution of order three, is an important tool: the FBST is used to test the hypotheses defined on the mixture weights space. Another feature of the paper is that all three distributions are reparametrized in that all the six parameter…
Computational convenience has led to widespread use of Bayesian inference with vague or flat priors to analyze state-space models in ecology. Vague priors are claimed to be objective and to let the data speak. Neither of these claims is valid. Statisticians have criticized the use of vague priors from philosophical to computational to pragmatic reasons. Ecologists, however, dismiss such criticisms as empty philosophical wonderings with no practical implications. We illustrate that use of vague priors in population viability analysis and occupancy models can have significant impact on the analy…
Bayesian joint modeling of longitudinal patient-reported outcomes and survival: an application to chronic obstructive pulmonary disease
Basque Center for Applied Mathematics BCAM（巴斯克应用数学中心）；University of the Basque Country UPV/EHU（巴斯克大学）；MRC Biostatistics Unit, University of Cambridge（剑桥大学医学研究委员会生物统计学单位）；IE University（IE大学）
This article proposes an efficient Bayesian inference for piecewise exponential hazard (PEH) models, which allow the effect of a covariate on the survival time to vary over time. The proposed inference methodology is based on a particle smoothing (PS) algorithm that depends on three particle filters. Efficient proposal (importance) distributions for the particle filters tailored to the nature of survival data and PEH models are developed using the Laplace approximation of the posterior distribution and linear Bayes theory. The algorithm is applied to both simulated and real data, and the resul…
Modelling time-varying and frequency-specific relationships between two brain signals is becoming an essential methodological tool to answer heoretical questions in experimental neuroscience. In this article, we propose to estimate a frequency Granger causality statistic that may vary in time in order to evaluate the functional connections between two brain regions during a task. We use for that purpose an adaptive Kalman filter type of estimator of a linear Gaussian vector autoregressive model with coefficients evolving over time. The estimation procedure is achieved through variational Bayes…
We study Bayesian linear regression models with skew-symmetric scale mixtures of normal error distributions. These kinds of models can be used to capture departures from the usual assumption of normality of the errors in terms of heavy tails and asymmetry. We propose a general non-informative prior structure for these regression models and show that the corresponding posterior distribution is proper under mild conditions. We extend these propriety results to cases where the response variables are censored. The latter scenario is of interest in the context of accelerated failure time models, wh…
The mean residual life function is a key functional for a survival distribution. It has a practically useful interpretation as the expected remaining lifetime given survival up to a particular time point, and it also characterizes the survival distribution. However, it has received limited attention in terms of inference methods under a probabilistic modeling framework. We seek to provide general inference methodology for mean residual life regression. Survival data often include a set of predictor variables for the survival response distribution, and in many cases it is natural to include the…
We introduce a numerically tractable formulation of Bayesian joint models for longitudinal and survival data. The longitudinal process is modelled using generalised linear mixed models, while the survival process is modelled using a parametric general hazard structure. The two processes are linked by sharing fixed and random effects, separating the effects that play a role at the time scale from those that affect the hazard scale. This strategy allows for the inclusion of non-linear and time-dependent effects while avoiding the need for numerical integration, which facilitates the implementati…
