Posterior Densities Of Parameters Research Articles

Simulation experiment design for calibration via active learning

Simulation models often have parameters as input and return outputs to understand the behavior of complex systems. Calibration is the process of estimating the values of the parameters in a simulation model in light of observed data from the system that is being simulated. When simulation models are expensive, emulators are built with simulation data as a computationally efficient approximation of an expensive model. An emulator then can be used to predict model outputs, instead of repeatedly running an expensive simulation model during the calibration process. Sequential design with an intelligent selection criterion can guide the process of collecting simulation data to build an emulator, making the calibration process more efficient and effective. This article proposes two novel criteria for sequentially acquiring new simulation data in an active learning setting by considering uncertainties on the posterior density of parameters. Analysis of several simulation experiments and real-data simulation experiments from epidemiology demonstrates that proposed approaches result in improved posterior and field predictions.

Bayesian analysis of Turkish Income and Living Conditions data, using clustered longitudinal ordinal modelling with Bridge distributed random effects

This article is motivated by the panel surveys, called Statistics on Income and Living Conditions (SILC), conducted annually on (randomly selected) country representative households to monitor EU 2020 aims on poverty reduction. We particularly consider the surveys conducted in Turkey within the scope of integration to the EU. Our main interests are on health aspects of economic and living conditions. The outcome is self-reported health that is clustered longitudinal ordinal, since repeated measures of it are nested within individuals and individuals are nested within families. Economic and living conditions have been measured through a number of individual- and family-level explanatory variables. The questions of interest are on the marginal relationships between the outcome and covariates that we address using a polytomous logistic regression with Bridge distributed random effects. This choice of distribution allows us to directly obtain marginal inferences in the presence of random effects. Widely used Normal distribution is also considered as the random effects distribution. Samples from the joint posterior densities of parameters and random effects are drawn using Markov Chain Monte Carlo. Interesting findings from the public health point of view are that differences were found between the subgroups of employment status, income level and panel year in terms of odds of reporting better health.

Robust Linear Filter with Parameter Estimation Under Student-t Measurement Distribution

In this paper, we propose an approximate Kalman filter of measurements following Student’s t-distribution by using the variational Bayes approach. This approach can decompose the estimation of multivariate parameters into a univariate estimation. The recursive formula for the approximate posterior densities of parameters and states is derived in detail. Then, the asymptotic Bayesian Cramer–Rao lower bounds are derived for the proposed filter. Numerical simulations verify both the performance of the proposed filter and the variance lower bounds under time-varying noise. The efficiency of the proposed filter is also demonstrated in a real application, namely an integrated strapdown inertial navigation system/Doppler velocity log shipborne test for navigation.

A Bayesian Analysis of Unobserved Component Models Using Ox

This article details a Bayesian analysis of the Nile river flow data, using a simple state space model. This allows the article to concentrate on implementation issues surrounding this model. For this data set, Metropolis-Hastings and Gibbs sampling algorithms are implemented in the programming language Ox. The Markov chain Monte Carlo methods only provide output conditioned upon the full data set. For filtered output, conditioning only on past observations, the particle filter is introduced. The sampling methods are flexible, and this advantage is used to extend the model to incorporate a stochastic volatility process. The volatility changes both in the Nile data, and for comparison also in daily S&P 500 return data, are investigated. The posterior density of parameters and states is found to provide information on which elements of the model are easily identifiable, and which elements are estimated with less precision.

Creaming - and the Depletion of Resources: A Bayesian Data Analysis

This paper considers sampling in proportion to size from a partly unknown distribution. The applied context is the exploration for undiscovered resources, like oil accumulations in different deposits, where the most promising deposits are likely to be drilled first, based on some geologic size indicators (“creaming”). A Log-normal size model with exponentially decaying creaming factor turns out to have nice analytical features in this context, and fits well available data, as demonstrated in Lillestol and Sinding-Larsen (2017). This paper is a Bayesian follow-up, which provides posterior parameter densities and predictive densities of future discoveries, in the case of uninformative prior distributions. The theory is applied to the prediction of remaining petroleum accumulations to be found on the mature part of the Norwegian Continental Shelf.

Inference of reaction rate parameters based on summary statistics from experiments

We present the results of an application of Bayesian inference and maximum entropy methods for the estimation of the joint probability density for the Arrhenius rate parameters of the rate coefficient of the H2/O2-mechanism chain branching reaction H+O2→OH+O. Available published data is summary statistics in terms of nominal values and error bars of the rate coefficient of this reaction at a number of temperature values obtained from shock-tube experiments. Our approach relies on generating data, in this case OH concentration profiles, consistent with the given summary statistics, using Approximate Bayesian Computation methods and a Markov chain Monte Carlo procedure. The approach permits the forward propagation of parametric uncertainty through the computational model in a manner that is consistent with the published statistics. A consensus joint posterior on the parameters is obtained by pooling the posterior parameter densities given each consistent data set. To expedite this process, we construct efficient surrogates for the OH concentration using a combination of Padé and polynomial approximants. These surrogate models adequately represent forward model observables and their dependence on input parameters and are computationally efficient to allow their use in the Bayesian inference procedure. We also utilize Gauss–Hermite quadrature with Gaussian proposal probability density functions for moment computation resulting in orders of magnitude speedup in data likelihood evaluation. Despite the strong non-linearity in the model, the consistent data sets all result in nearly Gaussian conditional parameter probability density functions. The technique also accounts for nuisance parameters in the form of Arrhenius parameters of other rate coefficients with prescribed uncertainty. The resulting pooled parameter probability density function is propagated through stoichiometric hydrogen–air auto-ignition computations to illustrate the need to account for correlation among the Arrhenius rate parameters of one reaction and across rate parameters of different reactions.

Joint Bayesian Analysis of Parameters and States in Nonlinear non‐Gaussian State Space Models

SummaryWe propose a new methodology for designing flexible proposal densities for the joint posterior density of parameters and states in a nonlinear, non‐Gaussian state space model. We show that a highly efficient Bayesian procedure emerges when these proposal densities are used in an independent Metropolis–Hastings algorithm or in importance sampling. Our method provides a computationally more efficient alternative to several recently proposed algorithms. We present extensive simulation evidence for stochastic intensity and stochastic volatility models based on Ornstein–Uhlenbeck processes. For our empirical study, we analyse the performance of our methods for corporate default panel data and stock index returns. Copyright © 2016 John Wiley & Sons, Ltd.

Bayesian Inference for State Space Model with Panel Data

Abstract The present work explores panel data set-up in a Bayesian state space model. The conditional posterior densities of parameters are utilized to determine the marginal posterior densities using the Gibbs sampler. An efficient one step ahead predictive density mechanism is developed to further the state of art in prediction-based decision making.

Bayesian inference for state space model with panel data

The present work explores panel data set-up in a Bayesian state space model. The conditional posterior densities of parameters are utilized to determine the marginal posterior densities using the Gibbs sampler. An efficient one step ahead predictive density mechanism is developed to further the state of art in prediction based decision making.

A Comparison of Bayesian Methods for Uncertainty Analysis in Hydraulic and Hydrodynamic Modeling

AbstractWe evaluate and compare the performance of Bayesian Monte Carlo (BMC), Markov chain Monte Carlo (MCMC), and the Generalized Likelihood Uncertainty Estimation (GLUE) for uncertainty analysis in hydraulic and hydrodynamic modeling (HHM) studies. The methods are evaluated in a synthetic 1D wave routing exercise based on the diffusion wave model, and in a multidimensional hydrodynamic study based on the Environmental Fluid Dynamics Code to simulate estuarine circulation processes in Weeks Bay, Alabama. Results show that BMC and MCMC provide similar estimates of uncertainty. The posterior parameter densities computed by both methods are highly consistent, as well as the calibrated parameter estimates and uncertainty bounds. Although some studies suggest that MCMC is more efficient than BMC, our results did not show a clear difference between the performance of the two methods. This seems to be due to the low number of model parameters typically involved in HHM studies, and the use of the same likelihood function. In fact, for these studies, the implementation of BMC results simpler and provides similar results to MCMC. The results of GLUE are, on the other hand, less consistent to the results of BMC and MCMC in both applications. The posterior probability densities tend to be flat and similar to the uniform priors, which can result in calibrated parameter estimates centered in the parametric space.

A Bayesian framework for describing and predicting the stochastic demand of home care patients

Home care providers are complex structures which include medical, paramedical and social services delivered to patients at their domicile. High randomness affects the service delivery, mainly in terms of unplanned changes in patients’ conditions, which make the amount of required visits highly uncertain. Hence, each reliable and robust resource planning should include the estimation of the future demand for visits from the assisted patients. In this paper, we propose a Bayesian framework to represent the patients’ demand evolution along with the time and to predict it in future periods. Patients’ demand evolution is described by means of a generalized linear mixed model, whose posterior densities of parameters are obtained through Markov chain Monte Carlo simulation. Moreover, prediction of patients’ demands is given in terms of their posterior predictive probabilities. In the literature, the stochastic description of home care patients’ demand is only marginally addressed and no Bayesian approaches exist to the best of our knowledge. Results from the application to a relevant real case show the applicability of the proposed model in the practice and validate the approach, since parameter densities in accordance to clinical evidences and low prediction errors are found.

Joint Bayesian Analysis of Parameters and States in Nonlinear, Non-Gaussian State Space Models

We propose a new methodology for designing flexible proposal densities for the joint posterior density of parameters and states in a nonlinear, non-Gaussian state space model. We show that a highly efficient Bayesian procedure emerges when these proposal densities are used in an independent Metropolis-Hastings algorithm or in importance sampling. Our method provides a computationally more efficient alternative to several recently proposed algorithms. We present extensive simulation evidence for stochastic intensity and stochastic volatility models based on Ornstein-Uhlenbeck processes. For our empirical study, we analyse the performance of our methods for corporate default panel data and stock index returns.(This paper is an updated version of the paper that appeared earlier as Barra, I., Hoogerheide, L.F., Koopman, S.J., and Lucas, A. (2013) Joint Independent Metropolis-Hastings Methods for Nonlinear Non-Gaussian State Space Models. TI Discussion Paper 13-050/III. Amsterdam: Tinbergen Institute.)

Quantification of parameter and model uncertainty for shape memory alloy bending actuators

In this article, we employ a Bayesian framework to estimate parameter and model uncertainty for shape memory alloy bending actuators. The Bayesian framework provides parameter densities, instead of ordinary least-squares optimal point estimates. Bayes’ rule relates a posterior parameter density to a prior density and likelihood. However, the posterior density is difficult to calculate directly for high-dimensional parameter spaces. Markov chain Monte Carlo methods overcome this difficulty indirectly by creating a Markov chain whose stationary density is the posterior. In this article, we utilize the Delayed Rejection Adaptive Metropolis algorithm for estimating parameter uncertainty. The shape memory alloy bending actuator is modeled using the homogenized energy framework, a computationally efficient and accurate model for various transductive materials. The model is summarized, and techniques for estimating the heat transfer parameters are presented. An algorithmic approach to quantifying uncertainty is useful for numerous reasons. The anticipated use is to quantify uncertainty for robust control algorithms. Robust control is an area of considerable research for smart materials such as shape memory alloy; however, the source of uncertainty is rarely quantified. The methods employed here would greatly aid in the design of robust controllers.

Nested sampling for parameter inference in systems biology: application to an exemplar circadian model

BackgroundModel selection and parameter inference are complex problems that have yet to be fully addressed in systems biology. In contrast with parameter optimisation, parameter inference computes both the parameter means and their standard deviations (or full posterior distributions), thus yielding important information on the extent to which the data and the model topology constrain the inferred parameter values.ResultsWe report on the application of nested sampling, a statistical approach to computing the Bayesian evidence Z, to the inference of parameters, and the estimation of log Z in an established model of circadian rhythms. A ten-fold difference in the coefficient of variation between degradation and transcription parameters is demonstrated. We further show that the uncertainty remaining in the parameter values is reduced by the analysis of increasing numbers of circadian cycles of data, up to 4 cycles, but is unaffected by sampling the data more frequently. Novel algorithms for calculating the likelihood of a model, and a characterisation of the performance of the nested sampling algorithm are also reported. The methods we develop considerably improve the computational efficiency of the likelihood calculation, and of the exploratory step within nested sampling.ConclusionsWe have demonstrated in an exemplar circadian model that the estimates of posterior parameter densities (as summarised by parameter means and standard deviations) are influenced predominately by the length of the time series, becoming more narrowly constrained as the number of circadian cycles considered increases. We have also shown the utility of the coefficient of variation for discriminating between highly-constrained and less-well constrained parameters.

A non-degenerate Rao-Blackwellised particle filter for estimating static parameters in dynamical models

The particle filter (PF) has emerged as a powerful tool for solving nonlinear and/or non-Gaussian filtering problems. When some of the states enter the model linearly, this can be exploited by using particles only for the “nonlinear” states and employing conditional Kalman filters for the “linear” states; this leads to the Rao-Blackwellised particle filter (RBPF). However, it is well known that the PF fails when the state of the model contains some static parameter. This is true also for the RBPF, even if the static states are marginalised analytically by a Kalman filter. The reason is that the posterior density of the static states is computed conditioned on the nonlinear particle trajectories, which are bound to degenerate over time. To circumvent this problem, we propose a method for targeting the posterior parameter density, conditioned on just the current nonlinear state. This results in an RBPF-like method, capable of recursive identification of nonlinear dynamical models with affine parameter dependencies.

A Bayesian Analysis of Unobserved Component Models UsingOx

This article details a Bayesian analysis of the Nile river flow data, using a similar state space model as other articles in this volume. For this data set, Metropolis-Hastings and Gibbs sampling algorithms are implemented in the programming language Ox. These Markov chain Monte Carlo methods only provide output conditioned upon the full data set. For filtered output, conditioning only on past observations, the particle filter is introduced. The sampling methods are flexible, and this advantage is used to extend the model to incorporate a stochastic volatility process. The volatility changes both in the Nile data and also in daily S&P 500 return data are investigated. The posterior density of parameters and states is found to provide information on which elements of the model are easily identifiable, and which elements are estimated with less precision.

A Flexible Bayesian Framework for Modeling Haplotype Association with Disease, Allowing for Dominance Effects of the Underlying Causative Variants

Multilocus analysis of single-nucleotide-polymorphism (SNP) haplotypes may provide evidence of association with disease, even when the individual loci themselves do not. Haplotype-based methods are expected to outperform single-SNP analyses because (i) common genetic variation can be structured into haplotypes within blocks of strong linkage disequilibrium and (ii) the functional properties of a protein are determined by the linear sequence of amino acids corresponding to DNA variation on a haplotype. Here, I propose a flexible Bayesian framework for modeling haplotype association with disease in population-based studies of candidate genes or small candidate regions. I employ a Bayesian partition model to describe the correlation between marker-SNP haplotypes and causal variants at the underlying functional polymorphism(s). Under this model, haplotypes are clustered according to their similarity, in terms of marker-SNP allele matches, which is used as a proxy for recent shared ancestry. Haplotypes within a cluster are then assigned the same probability of carrying a causal variant at the functional polymorphism(s). In this way, I can account for the dominance effect of causal variants, here corresponding to any deviation from a multiplicative contribution to disease risk. The results of a detailed simulation study demonstrate that there is minimal cost associated with modeling these dominance effects, with substantial gains in power over haplotype-based methods that do not incorporate clustering and that assume a multiplicative model of disease risks.

A BAYESIAN APPROACH TO THE IDENTIFICATION OF PIECEWISE LINEAR OUTPUT ERROR MODELS

In this paper we develop an algorithm for the identification of piecewise linear output error models for the case where the discrete mode of the underlying hybrid system is not known. The presented algorithm is based on a Bayesian framework, i.e. unknown model parameters are treated as random variables and described with probability density functions. The identification problem is posed as a problem of computing the posterior parameter densities, given the prior densities and the observed data. A suboptimal identification algorithm is derived. Operation of the algorithm is demonstrated on an example.

On the Multivariate Probit Model for Exchangeable Binary Data with Covariates

This paper considers the use of a multivariate binomial probit model for the analysis of correlated exchangeable binary data. The model can naturally accommodate both cluster and individual level covariates, while keeping a fairly flexible intracluster association structure. We discuss Bayesian estimation when a sample of independent clusters of varying sizes are available, and show how Gibbs sampling may be used to derive the posterior densities of parameters. The methodology is illustrated with two examples: the first involves epidemiological data from a study of familial disease aggregation; the second uses teratological data from a developmental toxicity application.

Bayesian analysis of hierarchical poisson models with latent variables

A Bayesian analysis of count data using a hierarchical model with latent variables has been proposed. The model allows for fixed covariates as well as direct modeling of latent variables as a function of covariates which are considered as primary importance. A Gibbs sampling algorithm with adaptive rejection sampling method is suggested to find posterior densities of parameters and latent variables. The proposed method is applied to epileptic seizure data arising from a study of progabide as an adjuvant therapy for seizure episodes.