A bayesian analysis of beta testing

In this article, we define a model for fault detection during the beta testing phase of a software design project. Given sampled data, we illustrate how to estimate the failure rate and the number of faults in the software using Bayesian statistical methods with various different prior distributions. Secondly, given a suitable cost function, we also show how to optimise the duration of a further test period for each one of the prior distribution structures considered

A semi-parametric model for circular data based on mixtures of beta distributions

This paper introduces a new, semi-parametric model for circular data, based on mixtures of shifted, scaled, beta (SSB) densities. This model is more general than the Bernstein polynomial density model which is well known to provide good approximations to any density with finite support and it is shown that, as for the Bernstein polynomial model, the trigonometric moments of the SSB mixture model can all be derived. Two methods of fitting the SSB mixture model are considered. Firstly, a classical, maximum likelihood approach for fitting mixtures of a given number of SSB components is introduced. The Bayesian information criterion is then used for model selection. Secondly, a Bayesian approach using Gibbs sampling is considered. In this case, the number of mixture components is selected via an appropriate deviance information criterion. Both approaches are illustrated with real data sets and the results are compared with those obtained using Bernstein polynomials and mixtures of von Mises distributions.Circular data, Shifted, scaled, beta distribution; Mixture models, Bernstein polynomials

BAYESIAN INFERENCE FOR A SOFTWARE RELIABILITY MODEL USING METRICS INFORMATION.

In this paper, we are concerned with predicting the number of faults N and the time to next failure of a piece of software. Information in the form of software metrics data is used to estimate the prior distribution of N via a Poisson regression model. Given failure time data, and a well known model for software failures, we show how to sample the posterior distribution using Gibbs sampling, as implemented in the package "WinBugs". The approach is illustrated with a practical example.

BAYESIAN INFERENCE FOR THE HALF-NORMAL AND HALF-T DISTRIBUTIONS

In this article we consider approaches to Bayesian inference for the half-normal and half-t distributions. We show that a generalized version of the normal-gamma distribution is conjugate to the half-normal likelihood and give the moments of this new distribution. The bias and coverage of the Bayesian posterior mean estimator of the halfnormal location parameter are compared with those of maximum likelihood based estimators. Inference for the half-t distribution is performed using Gibbs sampling and model comparison is carried out using Bayes factors. A real data example is presented which demonstrates the fitting of the half-normal and half-t models.

Bayesian hierarchical modelling of bacteria growth

Bacterial growth models are commonly used in food safety. Such models permit the prediction of microbial safety and the shelf life of perishable foods. In this paper, we study the problem of modelling bacterial growth when we observe multiple experimental results under identical environmental conditions. We develop a hierarchical version of the Gompertz equation to take into account the possibility of replicated experiments and we show how it can be fitted using a fully Bayesian approach. This approach is illustrated using experimental data from Listeria monocytogenes growth and the results are compared with alternative models. Model selection is undertaken throughout using an appropriate version of the deviance information criterion and the posterior predictive loss criterion. Models are fitted using WinBUGS via R2WinBUGS.Predictive microbiology, Growth models, Gompertz curve, Bayesian hierarchical modelling

Non-parametric methods for circular-circular and circular-linear

We present a non-parametric approach for the estimation of the bivariate distribution of two circular variables and the modelling of the joint distribution of a circular and a linear variable. We combine nonparametric estimates of the marginal densities of the circular and linear components with the use of class of nonparametric copulas, known as empirical Bernstein copulas, to model the dependence structure. We derive the necessary conditions to obtain continuous distributions defined on the cylinder for the circular-linear model and on the torus for the circular-circular model. We illustrate these two approaches with two sets of real environmental dataBernstein polynomials, Circular distributions, Circular-Circular data, Circular-linear data, Copulas, Non-parametric estimation

Circular Bernstein polynomial distributions

This paper introduces a new non-parametric approach to the modeling of circular data, based on the use of Bernstein polynomial densities which generalizes the standard Bernstein polynomial model to account for the specific characteristics of circular data. It is shown that the trigonometric moments of the proposed circular Bernstein polynomial distribution can all be derived in closed form. We comment on how to fit the Bernstein polynomial density approximation to a sample of data and illustrate our approach with a real data example.Circular data, Non-parametric modeling, Bernstein polynomials

BAYESIAN ESTIMATION FOR THE M/G/1 QUEUE USING A PHASE TYPE APPROXIMATION

This article deals with Bayesian inference and prediction for M/G/1 queueing systems. The general service time density is approximated with a class of Erlang mixtures which are phase type distributions. Given this phase type approximation, an explicit evaluation of measures such as the stationary queue size, waiting time and busy period distributions can be obtained. Given arrival and service data, a Bayesian procedure based on reversible jump Markov Chain Monte Carlo methods is proposed to estimate system parameters and predictive distributions.

On identifiability of MAP processes

Two types of transitions can be found in the Markovian Arrival process or MAP: with and without arrivals. In transient transitions the chain jumps from one state to another with no arrival; in effective transitions, a single arrival occurs. We assume that in practice, only arrival times are observed in a MAP. This leads us to define and study the Effective Markovian Arrival process or E-MAP. In this work we define identifiability of MAPs in terms of equivalence between the corresponding E-MAPs and study conditions under which two sets of parameters induce identical laws for the observable process, in the case of 2 and 3-states MAP. We illustrate and discuss our results with examples.Batch Markovian Arrival process, Hidden Markov models, Identifiability problems

Non-identifiability of the two state Markovian Arrival process

In this paper we consider the problem of identifiability of the two-state Markovian Arrival process (MAP2). In particular, we show that the MAP2 is not identifiable and conditions are given under which two different sets of parameters, induce identical stationary laws for the observable process.Batch Markovian Arrival process, Markov Renewal process, Hidden Markov models, Identifiability problems