Bayesian analysis of elapsed times in continuous‐time Markov chains

On the use of panel data in stochastic frontier models with improper priors

We develop a Bayesian analysis based on two different Jeffreys priors for the Student-t regression model with unknown degrees of freedom. It is typically difficult to estimate the number of degrees of freedom: improper prior distributions may lead to improper posterior distributions, whereas proper prior distributions may dominate the analysis. We show that Bayesian analysis with either of the two considered Jeffreys priors provides a proper posterior distribution. Finally, we show that Bayesian estimators based on Jeffreys analysis compare favourably to other Bayesian estimators based on priors previously proposed in the literature.

BackgroundBayesian phylogenetic inference holds promise as an alternative to maximum likelihood, particularly for large molecular-sequence data sets. We have investigated the performance of Bayesian inference with empirical and simulated protein-sequence data under conditions of relative branch-length differences and model violation.ResultsWith empirical protein-sequence data, Bayesian posterior probabilities provide more-generous estimates of subtree reliability than does the nonparametric bootstrap combined with maximum likelihood inference, reaching 100% posterior probability at bootstrap proportions around 80%. With simulated 7-taxon protein-sequence datasets, Bayesian posterior probabilities are somewhat more generous than bootstrap proportions, but do not saturate. Compared with likelihood, Bayesian phylogenetic inference can be as or more robust to relative branch-length differences for datasets of this size, particularly when among-sites rate variation is modeled using a gamma distribution. When the (known) correct model was used to infer trees, Bayesian inference recovered the (known) correct tree in 100% of instances in which one or two branches were up to 20-fold longer than the others. At ratios more extreme than 20-fold, topological accuracy of reconstruction degraded only slowly when only one branch was of relatively greater length, but more rapidly when there were two such branches. Under an incorrect model of sequence change, inaccurate trees were sometimes observed at less extreme branch-length ratios, and (particularly for trees with single long branches) such trees tended to be more inaccurate. The effect of model violation on accuracy of reconstruction for trees with two long branches was more variable, but gamma-corrected Bayesian inference nonetheless yielded more-accurate trees than did either maximum likelihood or uncorrected Bayesian inference across the range of conditions we examined. Assuming an exponential Bayesian prior on branch lengths did not improve, and under certain extreme conditions significantly diminished, performance. The two topology-comparison metrics we employed, edit distance and Robinson-Foulds symmetric distance, yielded different but highly complementary measures of performance.ConclusionsOur results demonstrate that Bayesian inference can be relatively robust against biologically reasonable levels of relative branch-length differences and model violation, and thus may provide a promising alternative to maximum likelihood for inference of phylogenetic trees from protein-sequence data.

To evaluate the usefulness of the Bayesian method for hepatic fibrosis staging with magnetic resonance elastography (MRE). The sample of this retrospective study comprised patients with chronic liver disease (n = 309), in whom histopathological fibrosis staging and MRE using either a 1.5T (n = 214) or a 3T magnetic resonance imaging (MRI) system (n = 95) had been performed. The optimal cutoff stiffness value was determined and used to calculate the discrimination ability of fibrosis staging by the cutoff method. The Bayesian method calculated post-MRE probability of each fibrosis stage, yielding MRE-based fibrosis staging without a cutoff value as well as the confidence of staging. We compared the discrimination ability in all patients and in a subgroup of patients with high (≥90%) posterior probability. The discrimination ability for hepatic fibrosis staging was comparable between the Bayesian method and the cutoff method in all patients because the accuracy of staging with the Bayesian method and the cutoff method in all patients was not different (P = 1.0000). However, in patients with high posterior probability by the Bayesian method, the accuracy of staging with the Bayesian method was significantly improved compared with that of the cutoff method in all patients; for discriminating stage ≥F2 from F0-F1 (98.9% vs. 94.8%, P = 0.0069); for ≥F3 (99.6% vs. 92.6%, P < 0.0001); and for F4 (100% vs. 94.2%, P = 0.0002). The Bayesian method has a highly accurate discrimination ability for noninvasive hepatic fibrosis staging using MRE, if the posterior probability is high. 3 Technical Efficacy: Stage 2 J. MAGN. RESON. IMAGING 2017;46:375-382.

Bayesian statistical methods have recently been combined with conventional methods for fisheries stock assessment (e.g. catch-age analysis) to provide a conceptually elegant approach for providing fishery management advice under uncertainty. Uncertainties in the advice provided can be conveyed using posterior probability distributions (or “posteriors”) for the potential outcomes of each policy option. Posteriors can be estimated using data (e.g. catch-age data and relative abundance indices) for the fish population of interest and prior probability distributions for population model parameters (e.g. stock-recruit function parameters) based on data from similar fish populations. Despite growing interest, Bayesian methods remain accessible to relatively few. To increase the accessibility of these methods, the conceptual basis for Bayesian statistical estimation is reviewed and set in the context of fisheries stock assessment. The use of Bayesian methods is illustrated by fitting a logistic model to relative abundance indices for Namibian hake (Merlucius capensis and M. paradoxus) and presenting a decision analysis of alternative harvest policy options. Some alternative approaches are outlined for constructing prior and posterior probability distributions and some recent applications in fisheries. Some of the problems that can be encountered while implementing Bayesian methods are also discussed.

Development and comparison in uncertainty assessment based Bayesian modularization method in hydrological modeling

The aim of this paper is to introduce a new model of a financial asset prices distribution. It is known that the probability distribution of an asset prices or returns is unknown in reality. The general model of asset prices based on continuous time Markov chains is proposed. For this reason the interarrivals between two price states are approximated by mixture of exponential distributions. Numerical-analytic approach is used to obtain the probability distribution of asset prices. The developed software allows creating the space of an asset prices, the matrix of transition rates among states, a system of equations to find the steady state probabilities of price states and solves the system of equations by method of imbedded Markov chains.

Reproduction numbers, like the basic reproduction number R0, play an important role in the analysis and application of dynamic models, including contagion models and ecological population models. One difficulty in deriving these quantities is that they must be computed on a model-by-model basis, since it is typically impractical to obtain general reproduction number expressions applicable to a family of related models, especially if these are of different dimensions (i.e., differing numbers of state variables). For example, this is typically the case for SIR-type infectious disease models derived using the linear chain trick. Here we show how to find general reproduction number expressions for such model families (which vary in their number of state variables) using the next generation operator approach in conjunction with the generalized linear chain trick (GLCT). We further show how the GLCT enables modelers to draw insights from these results by leveraging theory and intuition from continuous time Markov chains (CTMCs) and their absorption time distributions (i.e., phase-type probability distributions). To do this, we first review the GLCT and other connections between mean-field ODE model assumptions, CTMCs, and phase-type distributions. We then apply this technique to find reproduction numbers for two sets of models: a family of generalized SEIRS models of arbitrary finite dimension, and a generalized family of finite dimensional predator-prey (Rosenzweig-MacArthur type) models. These results highlight the utility of the GLCT for the derivation and analysis of mean field ODE models, especially when used in conjunction with theory from CTMCs and their associated phase-type distributions.

Bayesian Analysis of the Pragmatic Airway Resuscitation Trial

Reference priors are theoretically attractive for the analysis of geostatistical data since they enable automatic Bayesian analysis and have desirable Bayesian and frequentist properties. But their use is hindered by computational hurdles that make their application in practice challenging. In this work, we derive a new class of default priors that approximate reference priors for the parameters of some Gaussian random fields. It is based on an approximation to the integrated likelihood of the covariance parameters derived from the spectral approximation of stationary random fields. This prior depends on the structure of the mean function and the spectral density of the model evaluated at a set of spectral points associated with an auxiliary regular grid. In addition to preserving the desirable Bayesian and frequentist properties, these approximate reference priors are more stable, and their computations are much less onerous than those of exact reference priors. Unlike exact reference priors, the marginal approximate reference prior of correlation parameter is always proper, regardless of the mean function or the smoothness of the correlation function. This property has important consequences for covariance model selection. An illustration comparing default Bayesian analyses is provided with a dataset of lead pollution in Galicia, Spain.

The Generalized Pareto (GP) and Generalized extreme value (GEV) distributions play an important role in extreme value analyses, as models for threshold excesses and block maxima respectively. For each of these distributions we consider Bayesian inference using "reference" prior distributions (in the general sense of priors constructed using formal rules) for the model parameters, specifically a Jeffreys prior, the maximal data information (MDI) prior and independent uniform priors on separate model parameters. We investigate the important issue of whether these improper priors lead to proper posterior distributions. We show that, in the GP and GEV cases, the MDI prior, unless modified, never yields a proper posterior and that in the GEV case this also applies to the Jeffreys prior. We also show that a sample size of three (four) is sufficient for independent uniform priors to yield a proper posterior distribution in the GP (GEV) case.

<p xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" dir="auto" id="d612827e85"> <b>Objectives:</b> The utility and feasibility of Neural network (NN) and Bayesian in PK-PD analysis were investigated. Recently, Hierarchical Deep Compartment Modeling (HDCM) workflow was shown to be able to quantify uncertainty of NN model predicted parameters and was successfully tested on synthetic data from 10 subjects [ <a class="xref-link" href="#r1">1</a>]. While conceptually appealing, this approach incurs prohibitive run time for typical clinical datasets due to the complexity of the high-dimensional Bayesian posterior distribution. <p xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" dir="auto" id="d612827e93">To enable the full covariate modeling in PK-PD analyses with practical run time, this research utilized flow-based generative deep learning [ <a class="xref-link" href="#r2">2</a>] in addition to NN and Bayesian inference. A novel approach was developed, which trains normalizing flows (NFs) [ <a class="xref-link" href="#r2">2</a>] to generate both samples and their associated probabilities from the posterior distribution. This approach allows breaking down Bayesian inference from a large dataset into manageable subtasks that can be completed sequentially while accumulating all information from the whole dataset. <p xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" dir="auto" id="d612827e101"> <b>Methods:</b> The workflow was implemented in Julia to perform PopPK/PD analysis on simulated dataset for a hypothetical small molecule drug with pharmacodynamic activity following stimulatory response function. The data included 302 subjects from phase 1 and phase 2 studies, with formulation, subject type, weight, gender, age as PK covariates and 3 PD covariates (PDcov1-3). <p xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" dir="auto" id="d612827e106">The covariate-PK/PD parameter relationships were modeled by NN. Loose priors and standard normal distributions were set for fixed, random effect and NN parameters. The initial analysis step processed one sub-dataset (first subtask). The No-U-Turn Sampler (NUTS) [ <a class="xref-link" href="#r3">3</a>] was used to draw samples from the intermediate posterior distribution. A NF [ <a class="xref-link" href="#r2">2</a>] was trained to establish a probability distribution from these samples. The NF-induced distribution became the prior distribution input for the second analysis step, with the likelihood evaluated only for the next sub-dataset. This iterative procedure was repeated until the whole dataset was analyzed. <p xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" dir="auto" id="d612827e114"> <b>Results:</b> A six-step RealNVP[ <a class="xref-link" href="#r4">4</a>] architecture using NN conditioners of 2 hidden layers, width 100, was adequate for carrying posterior information forward across 30 sub-datasets. NUTS produced well-mixing chains (R^ &lt; 1.01). Full covariate analysis testing was completed within 3 days on a 64-core compute node. Posterior predictive checks confirmed that predicted concentrations/PD responses based on the posterior distributions captured observed drug concentrations/responses. Forest plots generated from the NN-predicted covariate effects correctly identified covariate-parameter relationships consistent with NONMEM-based Stepwise Covariate Modeling approach [ <a class="xref-link" href="#r5">5</a>]. <p xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" dir="auto" id="d612827e125"> <b>Conclusions:</b> Bayesian updates enabled by normalizing flows was shown to be a promising direction for improving scalability of NN-based PopPK/PD analyses. The newly developed method adds flexibility to analysis planning, allowing the posterior distribution to be provisionally generated from existing data while waiting for additional clinical studies, from which new information can be readily incorporated.

Quantile regression is a regression method that modelling a relationship between quantile of variable response and one or more variable predictors. Quantile regression has advantages that linear regression does not have; it is robust against outliers and can model heteroscedasticity data. The parameters of quantile regression can be estimated using the Bayesian method. The Bayesian method is a data analysis tool derived based on the Bayesian inference principle. Bayesian inference is the process of studying data analysis inductively with the Bayes theorem. To estimate regression parameters with Bayesian inference, it is necessary to find the posterior distribution of the regression parameters where the posterior distribution is proportional to the product of the prior distribution and its likelihood function. Since the calculation of the posterior distribution analytically is difficult to do if more parameters are estimated, the Markov Chain Monte Carlo (MCMC) method is proposed. The use of the Bayesian method in quantile regression has advantages, namely the use of MCMC has the advantages of obtaining sample parameter values from an unknown posterior distribution, using computationally efficient, and easy to implement. Yu and Moyeed (2001) introduced Bayesian quantile regression using the likelihood function of errors with an Asymmetric Laplace Distribution (ALD) and found that minimizing parameter estimates in quantile regression is the same as maximizing the likelihood function of errors with an Asymmetric Laplace Distribution (ALD). The method used to estimate quantile regression parameters is Gibbs sampling from the ALD, which is a combination of the exponential and normal distributions. To find the parameters of the regression model by sampling the posterior distribution found in this thesis. The results obtained from Gibbs sampling are a sample sequence of estimated parameters. After obtaining the sample sequences, the sample lines are averaged to obtain an estimated regression parameter. The case study in this thesis discusses the effect of risk factors from motor vehicle insurance customers on the size of claims submitted by customers.

Missing data are frequently encountered in longitudinal clinical trials. To better monitor and understand the progress over time, one must handle the missing data appropriately and examine whether the missing data mechanism is ignorable or nonignorable. In this article, we develop a new probit model for longitudinal binary response data. It resolves a challenging issue for estimating the variance of the random effects, and substantially improves the convergence and mixing of the Gibbs sampling algorithm. We show that when improper uniform priors are specified for the regression coefficients of the joint multinomial model via a sequence of one-dimensional conditional distributions for the missing data indicators under nonignorable missingness, the joint posterior distribution is improper. A variation of Jeffreys prior is thus established as a remedy for the improper posterior distribution. In addition, an efficient Gibbs sampling algorithm is developed using a collapsing technique. Two model assessment criteria, the deviance information criterion (DIC) and the logarithm of the pseudomarginal likelihood (LPML), are used to guide the choices of prior specifications and to compare the models under different missing data mechanisms. We report on extensive simulations conducted to investigate the empirical performance of the proposed methods. The proposed methodology is further illustrated using data from an HIV prevention clinical trial.

This paper considers the statistical approach to model software degradation process from time series data of system attributes. We first develop the continuous-time Markov chain (CTMC) model to represent the degradation level of system. By combining the CTMC with system attributes distributions, a continuous-time hidden Markov model (CT-HMM) is proposed as the basic model to represent the degradation level of system. To estimate model parameters, we develop the EM algorithm for CT-HMM. The advantage of this modeling is that the estimated model is directly applied to existing CTMC-based software aging and rejuvenation models. In numerical experiments, we exhibit the performance of our method by simulated data and also demonstrate estimating the software degradation process with experimental data in MySQL database system.