Bayesian analysis of change-points in poisson processes

Published ArticleChange-point analysis deals with the situation where an abrupt change has possibly taken place in the underlying mechanism that generates random variables. In a parametric setting, this means a change in the parameters of the underlying distribution. The interest is in whether such a change has actually taken place, and if it has, at which point in time. Also, there may have been more than one change during the period of interest. Application of change-point analysis is wide, but is particularly relevant in finance, the environment and medicine. The violability of markets may change abruptly, the rate and intensity of natural phenomena may change, or the effect of treatments in clinical trails may be studied. The literature on change-point problems is, by now, enormous. In this study we consider only the so-called non-sequential or fixed sample size version, although an informal sequential procedure, which follows from Smith (1975), is a routine consequence. Still, literature is substantial and our focus is on a fully Bayesian parametric approach. Use of the Bayesian framework for inference with regard to the change-point dates to work by Chernoff and Zacks (1964). Smith (1975) presents the Bayesian formulation for a finite sequence of independent observations. See also Zacks (1983). In our study we will consider only Poisson sequences and will address four situations: 1) When it is assumed that there is exactly one change-point, and proper priors are used. This can be generalised to more than one change-point. If the number of change-points is fixed and known, improper priors are also valid as will be explained later. 2) When there is a fixed number of change-points, the Markov Chain Monte Carlo method of Chib (1998) is useful, especially for large samples and multiple changepoints This approach will be described and applied. 3) When the number of change-points is unknown, and we want posterior probability distributions of the number of change-points, only proper priors are valid for calculating Bayes factors. In the case when no prior information is available, improper priors will cause the Bayes factor to have an indeterminate constant. In this case we apply the Fractional Bayes factor method of O’Hagan (1995). 4) When the data consists of multiple sequences, it is called multi-path changepoint analysis, and the distribution from which the change-points are drawn is of interest. Here the posterior distributions of parameters are estimated by MCMC methods. All the techniques are illustrated using simulated and real data sets

Phosphorus supplementation to natural pasture grazing for beef cows in the Western Highveld region of South Africa

Simmentaler cows grazing Cymbopogon-Themeda veld in the western Highveld region of South Africa (1985/86-1989/90) received three levels of phosphorus (P) supplementation. Dicalcium phosphate content of the supplement was manipulated to give intakes of 8 (CS), 4 (HS) or 0 (ZS) g P/cow/day during summer. All cows received supplemental P (10 g/cow/day) during winter (1986-1989). In 1990, supplemental P (10 (CW), 5 (HW) or 0 (ZW) g P/cow/day) was given in a winter maintenance supplement (protein, energy and minerals). Rainfall was above average during the trial period. Reproductive performance was not influenced by P supplementation. Mean livemass of the CS group was greatest (p 0.05) by P supplementation. Cow mass was ffected by winter P supplementation (p < 0.05). Both CW and HW displayed improved (p < 0.01) condition scores and higher (p < 0.01) bone P content than ZW. Fatalities (4) occurred in ZW due to P deficiency (74.5 mg P/cm 3 bone), and deficiency symptoms were manifested in the entire group. Both summer and winter veld is deficient in P, which makes continuous supplementation a recommended practice in this area. (South African Journal of Animal Science, 2000, 30(1): 43-52

Abordagem Bayesiana da curva de lactação de cabras Saanen de primeira e segunda ordem de parto Bayesian approach in the lactation curve of Saanen goats from first and second calving orders

O objetivo deste trabalho foi utilizar o método Bayesiano no ajuste do modelo de Wood a dados de produção de leite de cabras da raça Saanen. Dois grupos de animais da primeira e segunda lactação foram considerados. Amostras das distribuições marginais a posteriori dos parâmetros do modelo de Wood e das funções de produção derivadas desses parâmetros - pico de produção, tempo do pico de produção, persistência e produção total de leite - foram obtidas pelo algoritmo Gibbs Sampler. As inferências foram feitas em cada população e os resultados mostraram diferenças na taxa de decréscimo da produção após o pico e na persistência, indicando maior produção nos animais de segunda lactação. Realizou-se um estudo de simulação de dados para avaliar o método Bayesiano sob diferentes estruturas de matrizes de covariâncias dos parâmetros. Os resultados desse estudo indicam que o método é eficiente no estudo das curvas de lactação quando a matriz de covariância apresenta alta correlação dos parâmetros.<br>The objective of this work was to use the Bayesian method in the fitting of the Wood&acute;s model for milk production of Saanen goats. Two groups of animals from first and second lactation were considered in the analysis. The posterior marginal distributions for each parameter and production functions, peak milk yield, time of peak yield, persistency and total milk production, were obtained via Gibbs Sampler algorithm. The inference was done for each population. The results showed differences in the slope of the curve after the peak and in persistency, indicating highest production for the second lactation. The data were simulated for evaluating Bayesian method under several covariance matrices structures. The simulation results indicate the efficiency of this method for lactation curves studies when the covariance matrices show high correlation for parameters