Bayesian Prior Distribution Research Articles

Statistical application of barycentric rational interpolants: an alternative to splines

Spline curves, originally developed by numerical analysts for interpolation, are widely used in statistical work, mainly as regression splines and smoothing splines. Barycentric rational interpolants have recently been developed by numerical analysts, but have yet seen very few statistical applications. We give the necesssary information to enable the reader to use barycentric rational interpolants, including a suggestion for a Bayesian prior distribution, and explore the possible statistical use of barycentric interpolants as an alternative to splines. We give the all the necessary formulae, compare the numerical accuracy to splines for some Monte-Carlo datasets, and apply both regression splines and barycentric interpolants to two real datasets. We also discuss the application of these interpolants to data smoothing, where smoothing splines would normally be used, and exemplify the use of smoothing interpolants with another real dataset. Our conclusion is that barycentric interpolants are as accurate as splines, and no more difficult to understand and program. They offer a viable alternative methodology.

Species sampling models: consistency for the number of species

This paper considers species sampling models using constructions that arise from Bayesian nonparametric prior distributions. A discrete random measure, used to generate a species sampling model, can have either a countable infinite number of atoms, which has been the emphasis in the recent literature, or a finite number of atoms K, while allowing K to be assigned a prior probability distribution on the positive integers. It is the latter class of model we consider here, due to the interpretation of K as the number of species. We demonstrate the consistency of the posterior distribution of K as the sample size increases.

Evolutionary inference for function-valued traits: Gaussian process regression on phylogenies.

Biological data objects often have both of the following features: (i) they are functions rather than single numbers or vectors, and (ii) they are correlated owing to phylogenetic relationships. In this paper, we give a flexible statistical model for such data, by combining assumptions from phylogenetics with Gaussian processes. We describe its use as a non-parametric Bayesian prior distribution, both for prediction (placing posterior distributions on ancestral functions) and model selection (comparing rates of evolution across a phylogeny, or identifying the most likely phylogenies consistent with the observed data). Our work is integrative, extending the popular phylogenetic Brownian motion and Ornstein-Uhlenbeck models to functional data and Bayesian inference, and extending Gaussian process regression to phylogenies. We provide a brief illustration of the application of our method.

Consistent High-Dimensional Bayesian Variable Selection via Penalized Credible Regions

For high-dimensional data, particularly when the number of predictors greatly exceeds the sample size, selection of relevant predictors for regression is a challenging problem. Methods such as sure screening, forward selection, or penalized regressions are commonly used. Bayesian variable selection methods place prior distributions on the parameters along with a prior over model space, or equivalently, a mixture prior on the parameters having mass at zero. Since exhaustive enumeration is not feasible, posterior model probabilities are often obtained via long Markov chain Monte Carlo (MCMC) runs. The chosen model can depend heavily on various choices for priors and also posterior thresholds. Alternatively, we propose a conjugate prior only on the full model parameters and use sparse solutions within posterior credible regions to perform selection. These posterior credible regions often have closed-form representations, and it is shown that these sparse solutions can be computed via existing algorithms. The approach is shown to outperform common methods in the high-dimensional setting, particularly under correlation. By searching for a sparse solution within a joint credible region, consistent model selection is established. Furthermore, it is shown that, under certain conditions, the use of marginal credible intervals can give consistent selection up to the case where the dimension grows exponentially in the sample size. The proposed approach successfully accomplishes variable selection in the high-dimensional setting, while avoiding pitfalls that plague typical Bayesian variable selection methods.

Bayes Estimation under Conjugate Prior for the Case of Power Function Distribution

The Bayesian estimation approach is a non-classical device in the estimation part of statistical inference which is very useful in real world situation. The main objective of this paper is to study the Bayes estimators of the parameter of Power function distribution. In Bayesian estimation loss function, prior distribution and posterior distribution are the most important ingredients. In real life we try to minimize the loss and want to know some prior information about the problem to solve it accurately. The well known conjugate priors are considered for finding the Bayes estimator. In our study we have used different symmetric and asymmetric loss functions such as squared error loss function, quadratic loss function, modified linear exponential (MLINEX) loss function and non-linear exponential (NLINEX) loss function. The performance of the obtained estimators for different types of loss functions are then compared among themselves as well as with the classical maximum likelihood estimator (MLE). Mean Square Error (MSE) of the estimators are also computed and presented in graphs.

Spawner-Recruit Relationships of Demersal Marine Fishes: Prior Distribution of Steepness

Stock assessments use spawner-recruit functions to relate the reproductive capacity of a stock (e.g., total fecundity) to subsequent recruitment. The Beverton-Holt spawner-recruit function, perhaps the most widely used, is conventionally parameterized using a parameter that describes the stock's productivity. This parameter highly influences predicted population dynamics and responses to exploitation. Unfortunately, steepness can also be difficult to estimate reliably from data typical of stock assessments. In such cases, estimation can be improved by drawing inference from other stocks with similar life-history patterns. In particular, Bayesian prior distributions can formally be incorporated into stock assessments to inform estimation of steepness. In the present study, we used a meta-analytic approach to compute a prior distribution of steepness, focusing on marine demersal fishes. We similarly computed a prior distribution of maximum lifetime reproductive rate, a parameter inextricably related to steepness. In addition, we tested relationships between steepness and two life-history parameters linked to longevity-natural mortality and age at maturity-to examine the common assumption that long-lived, K-selected species have lower steepness values. In neither case was steepness significantly related to the life-history parameter. Our results should be directly applicable in stock assessments that apply the Beverton-Holt (or Ricker) function to marine demersal fishes, such as reef-associated species of the southeast United States in Atlantic, Caribbean, and Gulf of Mexico waters.

PRICING OF MEDICAL DEVICES UNDER COVERAGE UNCERTAINTY—A MODELLING APPROACH

Product vendors and manufacturers are increasingly aware that purchasers of health care will fund new clinical treatments only if they are perceived to deliver value-for-money. This influences companies' internal commercial decisions, including the price they set for their products. Other things being equal, there is a price threshold, which is the maximum price at which the device will be funded and which, if its value were known, would play a central role in price determination. This paper examines the problem of pricing a medical device from the vendor's point of view in the presence of uncertainty about what the price threshold will be. A formal solution is obtained by maximising the expected value of the net revenue function, assuming a Bayesian prior distribution for the price threshold. A least admissible price is identified. The model can also be used as a tool for analysing proposed pricing policies when no formal prior specification of uncertainty is available.

The Knowledge-Gradient Algorithm for Sequencing Experiments in Drug Discovery

We present a new technique for adaptively choosing the sequence of molecular compounds to test in drug discovery. Beginning with a base compound, we consider the problem of searching for a chemical derivative of the molecule that best treats a given disease. The problem of choosing molecules to test to maximize the expected quality of the best compound discovered may be formulated mathematically as a ranking-and-selection problem in which each molecule is an alternative. We apply a recently developed algorithm, known as the knowledge-gradient algorithm, that uses correlations in our Bayesian prior distribution between the performance of different alternatives (molecules) to dramatically reduce the number of molecular tests required, but it has heavy computational requirements that limit the number of possible alternatives to a few thousand. We develop computational improvements that allow the knowledge-gradient method to consider much larger sets of alternatives, and we demonstrate the method on a problem with 87,120 alternatives.

Model-Based Bayesian Adaptive Dose-Finding Designs for a Phase II Trial

We describe the planning of a dose-finding study for a compound currently in early Phase II clinical development. Data from a trial with the same primary endpoint are available for a marketed drug that has the same pharmacological mechanism, which provides strong prior information for some characteristics of the new compound. We develop and evaluate informative model-based Bayesian analyses. We also evaluate adaptive designs that change the allocation of patients to doses after an interim analysis. The performance of the model-based Bayesian analyses applied to the adaptive designs is compared to the performance of corresponding analyses applied to a nonadaptive design. The performance is also compared to model-based maximum likelihood estimation and simple pairwise comparison of treatment group means applied to the nonadaptive design to assess the contributions of the model, Bayesian prior distribution, and adaptive designs to improvements in decision and estimation performance.

Optimization of Maximum Entropy Model of Bayesian Prior Distribution Based on HPSO

A hybrid particle swarm optimization (HPSO) approach is proposed to solve the optimization problem of the maximum entropy model oriented to Bayesian prior distribution. HPSO introduces chaos mechanism to create better initial species population, and the power-function carrier is adopted to improve the ergodicity and the sufficiency of the chaos mechanism. Then HPSO uses an inertia weight, which can balance global and local searching capability and fasten convergence speed. A nonlinear constrained optimization model of prior distribution based on the principle of maximum entropy is set up. By using Lagrange multiplier this constrained optimization problem is transformed to a non-constrained optimal one, which is solved by PSO and HPSO algorithm. The simulation example shows that HPSO not only has a better performance at the aspect of solution precision but also converges more quickly.

Applying the Principle of Maximum Entropy in Bayesian Prior Distribution Assignment

Under the prior information that upper and lower bounds of the random quantity are symmetric with respect to the best estimate, this paper analyses the Bayesian prior distribution assignment using the principle of maximum entropy. With the exact lower and upper bounds, it approves uniform for the probability density function of the quantity and it has a curvilinear trapezoidal form for the inexact lower and upper bounds.

ASSESSMENT OF STRUCTURES SUBJECTED TO ACCIDENTAL ACTIONS USING CRISP AND UNCERTAIN FRAGILITY FUNCTIONS

An application of fragility functions to the assessment of potential damage due to an accidental action is analysed. The assessment is carried out as an estimation of the probability of a foreseeable damage event (damage probability). This probability is expressed as a mean value of a fragility function developed for the damage event under study. A Bayesian prior (posterior) distribution specified for this mean value is used as an estimate of the damage probability. The prior distribution is derived by transforming prior knowledge through the fragility function and “mapping” this knowledge on the scale of probability values. The technique of Bayesian bootstrap resampling is applied to update the prior distribution. The new information used for the updating consists of a relatively small number of experimental observations of the accidental action. To facilitate the updating, these observations are transformed into a fictitious statistical sample of fragility function values. The updating is first carried out with a fragility function which expresses aleatory uncertainty only. Then it is proposed how to perform the updating with the fragility function which quantifies both aleatory and epistemic uncertainty. This is done by discretising continuous distributions of the epistemic uncertainty related to values (parameters) of the fragility function. The proposed approach allows to utilise different sources of information for the damage assessment. A potential field of application of this approach is risk studies of hazardous industrial facilities. Santrauka Analizuojamas pažeidžiamumo funkcijų taikymas vertinant potencialius statybinių konstrukcijų pažeidimus avariniais poveikiais. Vertinimas atliekamas skaičiuojant galimos konstrukcijos pažaidos tikimybę. Ši tikimybė yra išreiškiama vidutine pažeidžiamumo funkcijos reikšme. Ta funkcija yra formuojama analizuojamam pažaidos įvykiui. Apriorinis ir aposteriorinis Bajeso skirstiniai yra taikomi pažaidos tikimybės reikšmei vertinti. Apriorinis skirstinys yra gaunamas pasinaudojant turima informacija apie avarinį poveikį ir transformuojant šią informaciją per pažeidžiamumo funkciją. Aposteriorinis skirstinys yra gaunamas pasitelkiant naują, eksperimentinę informaciją apie avarinį poveikį. Aposterioriniam skirstiniui gauti taikomas kartotinio statistinio ėmimo (būtstrapo) metodas. Naują informaciją sudaro eksperimentiniai avarinio poveikio charakteristikų matavimai, kurie tiksliai atitinka konstrukcijos ekspozicijos tiriamo poveikio situaciją. Apriorinis ir aposteriorinis skirstiniai išreiškia episteminį neapibrėžtumą vertinamos pažaidos tikimybės reikšmės atžvilgiu. Šie skirstiniai yra gaunami taikant tiek pažeidžiamumo funkciją, kuri išreiškia tik stochastinį neapibrėžtumą, tiek funkciją, kurios reikšmės yra neapibrėžtos epistemine prasme. Potenciali siūlomo metodo taikymo sritis yra pavojingų pramoninių objektų rizikos vertinimas.

Bayesian Reasoning in Managerial Decisions on the Choice of Equipment for the Prevention of Industrial Accidents

The managerial problem of the choice among alternative protective equipment used to prevent industrial (technological) accidents is considered. The choice takes into account potential failures of the equipment or, conversely, the equipment reliability. Such failures can substantiate contributors to escalations of industrial accidents. The problem of the choice is formulated in the form of a multi-attribute selection. The probability of failure of alternative equipment sets is used as an attribute of the selection problem. An estimation of this probability by means of Bayesian statistical theory is considered. Bayesian prior and posterior distributions are applied as an estimate of failure probability. These distributions are incorporated in the selection problem as uncertain attributes. Development of prior distributions of individual alternatives is discussed in detail. The prior and posterior distributions are treated as measures of the epistemic uncertainty (state-of-knowledge) related to unknown values of failure probabilities. The modelling of uncertainty related to the failure probability corresponds to the classical Bayesian approach to risk assessment. It is suggested to apply the uncertain failure probabilities to developing risk profile for each of the alternative equipments and to incorporate the risk profile to the multi-attribute decision making. The epistemic uncertainty in the failure probabilities and elements of risk profile is quantified by means of Bayesian statistical theory. It is shown that this uncertainty can be reduced by applying Bayesian updating procedure when new data on equipment failures is obtained. Probable cases of the application of this procedure are discussed. This discussion relates the acquisition of the new statistical evidence used for Bayesian updating to the moments, at which managerial decisions are made. It is suggested to solve the problem of the multiattribute selection with uncertain attributes by means of uncertainty propagation. This propagation is accomplished by means of Monte Carlo simulation. The epistemic uncertainty related to the attributes is transformed into a discrete distribution of epistemic uncertainty. Probability masses of this distribution express the chance of individual alternatives to be selected as the best one. The result of this selection will be the alternative with the largest epistemic weight. The potential field of the application of the proposed approach is the management of technological risks present in many industrial facilities and non-industrial installations (dwellings, offices, public places) which are subjected to the hazard of accidents. The approach proposed in the paper allows to make managerial decisions which take into account the reliability of protective equipment. In addition, this approach will allow to utilize data on equipment failures encountered in the past. Such data is usually scarce and expensive. The Bayesian framework is best suited for the application of such data.

Probabilistic Applications of the Schlömilch Transformation

The Schlömilch transformation, long used by mathematicians for integral evaluation, allows probability mass to be redistributed, thus transforming old distributions to new ones. The transformation is used to introduce some new families of distributions on ℛ+. Their general properties are studied, i.e., distributional shape and skewness, moments and inverse moments, hazard function, and random number generation. In general, these distributions are suitable for modeling data where the hazard function initially rises steeply. Their usefulness is illustrated by fitting some human weight data. Besides data fitting, one possible use of the new distributions could be in sensitivity or robustness studies, for example as Bayesian prior distributions.

Pooled analyses of effects on visual and visuomotor performance from exposure to magnetic stray fields from MRI scanners: Application of the Bayesian framework

To pool measurement data from individual studies on acute and temporal neurobehavioral effects of stray fields and re-analyze these using a Bayesian framework. Data from tests assessing effects of exposure to stray fields (<1600 mT) from MRI systems (1.5-7.0 T) on visuomotor and visual sensory systems collected in three relatively small case-crossover studies of volunteers seated in the magnet stray field were analyzed together using hierarchical regression models. Bayesian prior distributions were specified such that a priori an association with electromagnetic field (EMF) exposure was absent, and were updated with measurement data into posterior distributions. The posterior distributions suggested that visuomotor speed, but not precision, was affected by exposure (-0.2% to -0.7% per 100 mT). The visual contrast threshold at stronger contrasts also increased with increased EMF exposure (-1%/100 mT). Using a Bayesian framework with conservative priors appeared to be an effective technique to assess subtle effects of exposure to stray magnetic fields. The posterior distributions were dominated by the observed data, which provides additional compelling evidence that the visuomotor domain and the visual contrast threshold level are negatively affected in the presence of stray fields of 1.5-T to 7.0-T MRI systems.

Modeling payback from research into the efficacy of left-ventricular assist devices as destination therapy

Ongoing developments in design have improved the outlook for left-ventricular assist device (LVAD) implantation as a therapy in end-stage heart failure. Nevertheless, early cost-effectiveness assessments, based on first-generation devices, have not been encouraging. Against this background, we set out (i) to examine the survival benefit that LVADs would need to generate before they could be deemed cost-effective; (ii) to provide insight into the likelihood that this benefit will be achieved; and (iii) from the perspective of a healthcare provider, to assess the value of discovering the actual size of this benefit by means of a Bayesian value of information analysis. Cost-effectiveness assessments are made from the perspective of the healthcare provider, using current UK norms for the value of a quality-adjusted life-year (QALY). The treatment model is grounded in published analyses of the Randomized Evaluation of Mechanical Assistance for the Treatment of Congestive Heart Failure (REMATCH) trial of first-generation LVADs, translated into a UK cost setting. The prospects for patient survival with second-generation devices is assessed using Bayesian prior distributions, elicited from a group of leading clinicians in the field. Using established thresholds, cost-effectiveness probabilities under these priors are found to be low (approximately .2 percent) for devices costing as much as 60,000 pounds. Sensitivity of the conclusions to both device cost and QALY valuation is examined. In the event that the price of the device in use would reduce to 40,000 pounds, the value of the survival information can readily justify investment in further trials.

Bayesian Methods for Highly Correlated Exposure Data

Studies that include individuals with multiple highly correlated exposures are common in epidemiology. Because standard maximum likelihood techniques often fail to converge in such instances, hierarchical regression methods have seen increasing use. Bayesian hierarchical regression places prior distributions on exposure-specific regression coefficients to stabilize estimation and incorporate prior knowledge, if available. A common parametric approach in epidemiology is to treat the prior mean and variance as fixed constants. An alternative parametric approach is to place distributions on the prior mean and variance to allow the data to help inform their values. As a more flexible semiparametric option, one can place an unknown distribution on the coefficients that simultaneously clusters exposures into groups using a Dirichlet process prior. We also present a semiparametric model with a variable-selection prior to allow clustering of coefficients at 0. We compare these 4 hierarchical regression methods and demonstrate their application in an example estimating the association of herbicides with retinal degeneration among wives of pesticide applicators.

ACCIDENTAL EXPLOSIONS ON THE RAILWAY: SIMULATION‐BASED PREDICTION OF DAMAGE TO NEARBY BUILDINGS

A procedure for estimating potential damage to buildings induced by accidental explosions on the railway is developed. By the damage failures of nearby structures due to actions generated by the accidental explosions are meant. This damage is measured in terms of probabilities of potential failures caused by explosions. The estimation of the damage probabilities is based on stochastic simulation of railway accidents involving an explosion. The proposed simulation‐based procedure quantifies epistemic (state‐of‐knowledge) uncertainties in the damage probabilities. These uncertainties are expressed in terms of Bayesian prior and posterior distributions. The foundation of the procedure is a computer intensive method known as the Bayesian bootstrap. It is used for approximating the posterior distributions of damage probabilities. The application of the Bayesian bootstrap makes the proposed procedure highly automatic and convenient for assessing structures subjected to the hazard of the accidental actions. In addition, it can be used for specifying safe distances between the railway and nearby buildings. Structures of these buildings can be designed for tolerable probabilities of failures induced by accidental explosions.

Propagation of Population Pharmacokinetic Information Using a Bayesian Approach: Comparison with Meta-Analysis

We investigated the propagation of population pharmacokinetic information across clinical studies by applying Bayesian techniques. The aim was to summarize the population pharmacokinetic estimates of a study in appropriate statistical distributions in order to use them as Bayesian priors in consequent population pharmacokinetic analyses. Various data sets of simulated and real clinical data were fitted with WinBUGS, with and without informative priors. The posterior estimates of fittings with non-informative priors were used to build parametric informative priors and the whole procedure was carried on in a consecutive manner. The posterior distributions of the fittings with informative priors where compared to those of the meta-analysis fittings of the respective combinations of data sets. Good agreement was found, for the simulated and experimental datasets when the populations were exchangeable, with the posterior distribution from the fittings with the prior to be nearly identical to the ones estimated with meta-analysis. However, when populations were not exchangeble an alternative parametric form for the prior, the natural conjugate prior, had to be used in order to have consistent results. In conclusion, the results of a population pharmacokinetic analysis may be summarized in Bayesian prior distributions that can be used consecutively with other analyses. The procedure is an alternative to meta-analysis and gives comparable results. It has the advantage that it is faster than the meta-analysis, due to the large datasets used with the latter and can be performed when the data included in the prior are not actually available.

Estimating in Vitro Mitochondrial Oxygen Consumption During Muscle Contraction and Recovery: A Novel Approach that Accounts for Diffusion

A deconvolution algorithm, based on a Bayesian statistical framework and smoothing spline technique, is applied to reconstructing input functions from noisy measurements in biological systems. Deconvolution is usually ill-posed. However, placing a Bayesian prior distribution on the input function can make the problem well-posed. Using this algorithm and a computational model of diffusional oxygen transport in an approximately cylindrical muscle (about 0.5-mm diameter and 10-mm long mouse leg muscle), the time course of muscle oxygen uptake and mitochondrial oxygen consumption, both during isometric twitch contractions (at various frequencies) and the recovery period, is estimated from polarographic measurements of oxygen concentration on the muscle surface. An important feature of our experimental protocol is the availability of data for the apparatus characteristics. From these time courses, the actual mitochondrial consumption rates during resting and exercise states can be estimated. Mitochondrial oxygen consumption rate increased during stimulation to a maximum steady state value approximately five times of the resting value of 0.63 nmol/s/g wet weight for the stimulation conditions studied. Diffusion slowed the kinetic responses to the contraction but not the steady state fluxes during the stimulation interval.