Bayesian decision-making in inventory modelling

The objective of this paper is to advocate the use of Bayesianmethods in tackling decision problems with limited past data.It is assumed that a Bayesian approach is least likely to besuccessful when there is no information on which to base a meaningfulprior. Here we use a limiting, invariant, form of the conjugateprior distribution to represent this ignorance. The resultsof decisions based on Bayesian methods with this ‘non-informative’prior are compared with those which result from deriving a pointestimate for the unknown parameter. The particular context consideredhere is that of a single-period inventory model with compoundPoisson demand made up of a known demand size distribution butan unknown demand rate. The demand rate is assumed to be highenough for a normal approximation to the compound Poisson distributionto be used, in which case it is possible to analyse the behaviourdirectly. An extension to the multi-period model with zero leadtime is considered briefly. The results lend support to theuse of Bayesian methods, with or without a meaningful prior,for which the analysis and computation are no more complex thanthose required by standard methods.     

On computational aspects of Bayesian spatial models: influence of the neighboring structure in the efficiency of MCMC algorithms

This study applies computationally intensive methods for Bayesian analysis of spatially distributed data. It is assumed that the space is divided in contiguous and disjoint regions or areas. The neighboring structure in a given problem may indicate a wide range of number of neighbors per area, ranging from very few neighbors to cases where all areas neighbor each other. The main aim of this work is to evaluate the influence of neighborhood on results of Markov Chain Monte Carlo (MCMC) methods. Proper and improper prior specifications for state parameters are compared. Three schemes, proposed in the literature, for sampling from the joint posterior distribution are also compared. The comparison criterion is based on the autocorrelation structure of the chains. Two classes of models are studied: the first one is characterized by a simple model without any explanatory variables and the second one is an extension with multiple regression components. Initially, sensitivity of the analysis to different prior distributions is addressed. Finally, extensive empirical analyses confront the outcomes obtained with different neighboring arrangements of the units. Results are shown to generalize those obtained with dynamic or state space models.     

Optimal selection from distributions with unknown parameters: Robustness of Bayesian models

We consider the problem of making one choice from a known number of i.i.d. alternatives. It is assumed that the distribution of the alternatives has some unknown parameter. We follow a Bayesian approach to maximize the discounted expected value of the chosen alternative minus the costs for the observations. For the case of gamma and normal distribution we investigate the sensitivity of the solution with respect to the prior distributions. Our main objective is to derive monotonicity and continuity results for the dependence on parameters of the prior distributions. Thus we prove some sort of Bayesian robustness of the model.     

Minimax pointwise estimation of an anisotropic regression function with unknown density of the design

Our aim in this paper is to estimate with best possible accuracy an unknown multidimensional regression function at a given point where the design density is also unknown. To reach this goal, we will follow the minimax approach: it will be assumed that the regression function belongs to a known anisotropic Hölder space. In contrast to the parameters defining the Hölder space, the density of the observations is assumed to be unknown and will be treated as a nuisance parameter. New minimax rates are exhibited as well as local polynomial estimators which achieve these rates. As these estimators depend on a tuning parameter, the problem of its selection is also discussed.     

Parameter estimation for a two-parameter bathtub-shaped lifetime distribution

A two-parameter distribution was revisited by Chen (2000) [7]. This distribution can have a bathtub-shaped or increasing failure rate function which enables it to fit real lifetime data sets. Maximum likelihood and Bayes estimates of the two unknown parameters are discussed in this paper. It is assumed in the Bayes case that the unknown parameters have gamma priors. Explicit forms of Bayes estimators cannot be obtained. Different approximations are used to establish point estimates and two sided Bayesian probability intervals for the parameters. Monte Carlo simulations are applied to the comparison between the maximum likelihood estimates and the approximate Bayes estimates obtained under non-informative prior assumptions. Analysis of a real data set is also been presented for illustrative purposes.     

Tracking the output of a poorly known markov process

Tracking the output of an unknown Markov process with unknown generator and unknown output function is considered. It is assumed the unknown quantities have a known prior probability distribution. It is shown that the optimal control is a linear feedback in the tracking error plus the conditional expectation of a quantity involving the unknown generator and output function of the Markov process. The results also have application to Bayesian identification of hidden Markov models     

Bayesian Binary Segmentation Procedure for a Poisson Process With Multiple Changepoints

We observe n events occurring in (0, T] taken from a Poisson process. The intensity function of the process is assumed to be a step function with multiple changepoints. This article proposes a Bayesian binary segmentation procedure for locating the changepoints and the associated heights of the intensity function. We conduct a sequence of nested hypothesis tests using the Bayes factor or the BIC approximation to the Bayes factor. At each comparison in the binary segmentation steps, we need only to compare a singlechangepoint model to a no-changepoint model. Therefore, this method circumvents the computational complexity we would normally face in problems with an unknown (large) number of dimensions. A simulation study and an analysis on a real dataset are given to illustrate our methods.     

Analysis of fluid storage tanks including foundation-superstructure interaction

An analytical formulation is developed to predict the flexural behavior of a cylindrical liquid storage tank resting on an isotropic elastic soil medium, which is modelled as a half space. The interface between the plate foundation and the soil medium is considered to be smooth and continuous. The plate deflection function is assumed in the form of a power series expansion in terms of the radial coordinate. The procedure accounts for the interactions between the tank wall and the plate foundation, and between the plate foundation and the soil medium. The principle of minimum potential energy is used to evaluate the unknown coefficients appearing in the assumed power series expansion and also the unknown interacting forces at the tank wall-plate foundation junction. Any number of terms can be considered in the assumed deflection function. Analytical expressions are obtained for the plate foundation deflections and radial moment, the contact stress distribution, the tank wall displacements, and the tank wall stress resultants. The results obtained compare well with the finite element analysis of a similar problem. Results of a parametric study are also presented to demonstrate the effect of the various geometric and material parameters on the flexural behavior of the system.     

Bayesian stopping rules for multistart global optimization methods

By far the most efficient methods for global optimization are based on starting a local optimization routine from an appropriate subset of uniformly distributed starting points. As the number of local optima is frequently unknown in advance, it is a crucial problem when to stop the sequence of sampling and searching. By viewing a set of observed minima as a sample from a generalized multinomial distribution whose cells correspond to the local optima of the objective function, we obtain the posterior distribution of the number of local optima and of the relative size of their regions of attraction. This information is used to construct sequential Bayesian stopping rules which find the optimal trade off between reliability and computational effort.     

Relationship between the posterior distributions of two regular Bayesian experiments related to a fixed family of sampling distributions

Let us consider twoRegular Bayesian Experiments (see [1]) related to a fixed family of sampling distributions in which the parameter space and the sample space are assumed to be Polish Spaces. In this paper we shall study the relationship between the posterior distributions of these two Bayesian Experiments considering all the different cases concerning the Lebesgue decomposition of the second prior distribution w.r.t. the first one.     

Bayesian Sparse Group Selection

This article proposes a Bayesian approach for the sparse group selection problem in the regression model. In this problem, the variables are partitioned into different groups. It is assumed that only a small number of groups are active for explaining the response variable, and it is further assumed that within each active group only a small number of variables are active. We adopt a Bayesian hierarchical formulation, where each candidate group is associated with a binary variable indicating whether the group is active or not. Within each group, each candidate variable is also associated with a binary indicator, too. Thus, the sparse group selection problem can be solved by sampling from the posterior distribution of the two layers of indicator variables. We adopt a group-wise Gibbs sampler for posterior sampling. We demonstrate the proposed method by simulation studies as well as real examples. The simulation results show that the proposed method performs better than the sparse group Lasso in terms of selecting the active groups as well as identifying the active variables within the selected groups. Supplementary materials for this article are available online.     

Optimal speed of detection in generalized Wiener disorder problems

We define a general Wiener disorder problem in which a sudden change in a time profile of unknown size has to be detected in white noise of small intensity. Since both the time of the change and its size are unknown, this problem is considerably harder than standard Wiener disorder problems where the size of the change is assumed to be known a priori. We formulate the problem within the Bayesian framework of nonlinear filtering theory, and use Strassen's functional law of the iterated logarithm to bound stochastic measures which arise in the nonlinear filtering equations. This leads to explicit expressions for the detection delay in the optimal statistics for small noise intensities, and we indicate how these can be used to analyse the detection delays of recursive suboptimal detection algorithms for this problem.     

Bayesian estimation of incompletely observed diffusions

We present a general framework for Bayesian estimation of incompletely observed multivariate diffusion processes. Observations are assumed to be discrete in time, noisy and incomplete. We assume the drift and diffusion coefficient depend on an unknown parameter. A data-augmentation algorithm for drawing from the posterior distribution is presented which is based on simulating diffusion bridges conditional on a noisy incomplete observation at an intermediate time. The dynamics of such filtered bridges are derived and it is shown how these can be simulated using a generalised version of the guided proposals introduced in Schauer, Van der Meulen and Van Zanten (2017, Bernoulli 23(4A)).     

Inference for an anisotropic diffusion model

The vector sum of a white noise in an unknown hyperspace and an Ornstein-Uhlenbeck process in an unknown line is observed through sharp linear test functions over a finite time span. The parameters associated with the white noise (including the hyperplane) are determinable with precision and index the measure-equivalence classes in the relevant sample space. An intraclass relative density provides a basis for Bayesian inference of the remaining parameters.     

A Bayesian approach for quantile and response probability estimation with applications to reliability

In this paper we propose a Bayesian approach for the estimation of a potency curve which is assumed to be nondecreasing and concave or convex. This is done by assigning the Dirichlet as a prior distribution for transformations of some unknown parameters. We motivate our choice of the prior and investigate several aspects of the problem, including the numerical implementation of the suggested scheme. An approach for estimating the quantiles is also given. By casting the problem in a more general context, we argue that distributions which are IHR or IHRA can also be estimated via the suggested procedure. A problem from a government laboratory serves as an example to illustrate the use of our procedure in a realistic scenario.     

Bayesian accelerated life test plans for series systems with Weibull component lifetimes

This article presents optimal Bayesian accelerated life test plans for series systems under Type-I censoring scheme. First, the component lifetimes are assumed to follow independent Weibull distributions. The scale parameters of Weibull lifetime distributions are related to the external stress variable through a general stress translation function. For a fixed number of design points, optimal Bayesian ALT plans are first obtained by solving constrained optimization problems under two different Bayesian design criteria. The global optimality of the resulting fixed-point optimal designs is then verified via the General Equivalence Theorem. This article also provides the optimized compromise ALT plans which are extremely useful in real-life applications. A detailed sensitivity analysis is then performed to find out the effect of various planning inputs on the resulting optimal Bayesian ALT plans. A simulation study is then conducted to visualize the resulting sampling variations from the optimal Bayesian ALT plans. Finally, this article considers a series system with dependent component lifetimes. Optimal ALT plans are obtained assuming a Gamma frailty model.     

Stress–strength reliability estimation under the standard two-sided power distribution

In this report, the distribution for setting up a system reliability exposed to some stress is studied. The standard two-sided power distribution is assumed to be the underlying distribution. We obtained the exact expressions and estimates for the reliability by applying different methods such as maximum likelihood and Bayesian estimators. Three different scenarios were examined: known and equal reflection parameters, known but unequal reflection parameters, and all parameters are unknown, providing practical guidance and recommendations for the estimator design. For large samples, we recommend use of the parametric bootstrap method with the maximum likelihood estimate. Real data sets were used to illustrate the performances of the estimators.     

Quadrature Methods for Bayesian Optimal Design of Experiments With Nonnormal Prior Distributions

Many optimal experimental designs depend on one or more unknown model parameters. In such cases, it is common to use Bayesian optimal design procedures to seek designs that perform well over an entire prior distribution of the unknown model parameter(s). Generally, Bayesian optimal design procedures are viewed as computationally intensive. This is because they require numerical integration techniques to approximate the Bayesian optimality criterion at hand. The most common numerical integration technique involves pseudo Monte Carlo draws from the prior distribution(s). For a good approximation of the Bayesian optimality criterion, a large number of pseudo Monte Carlo draws is required. This results in long computation times. As an alternative to the pseudo Monte Carlo approach, we propose using computationally efficient Gaussian quadrature techniques. Since, for normal prior distributions, suitable quadrature techniques have already been used in the context of optimal experimental design, we focus on quadrature techniques for nonnormal prior distributions. Such prior distributions are appropriate for variance components, correlation coefficients, and any other parameters that are strictly positive or have upper and lower bounds. In this article, we demonstrate the added value of the quadrature techniques we advocate by means of the Bayesian D-optimality criterion in the context of split-plot experiments, but we want to stress that the techniques can be applied to other optimality criteria and other types of experimental designs as well. Supplementary materials for this article are available online.     

Convergence Rates for Logspline Tomography

We consider bivariate logspline density estimation for tomography data. In the usual logspline density estimation for bivariate data, the logarithm of the unknown density function is estimated by tensor product splines, the unknown parameters of which are given by maximum likelihood. In this paper we use tensor product B-splines and the projection-slice theorem to construct the logspline density estimators for tomography data. Rates of convergence are established for log-density functions assumed to belong to a Besov space.     

A general approach to Bayesian portfolio optimization

We develop a general approach to portfolio optimization taking account of estimation risk and stylized facts of empirical finance. This is done within a Bayesian framework. The approximation of the posterior distribution of the unknown model parameters is based on a parallel tempering algorithm. The portfolio optimization is done using the first two moments of the predictive discrete asset return distribution. For illustration purposes we apply our method to empirical stock market data where daily asset log-returns are assumed to follow an orthogonal MGARCH process with t-distributed perturbations. Our results are compared with other portfolios suggested by popular optimization strategies.