1,721,041 research outputs found
Modelling trends in road accident frequency - Bayesian inference for rates with uncertain exposure
Full text linkBayesian inference for Poisson and multinomial log-linear models
Full text linkCategorical data frequently arise in applications in the social sciences. In such applications,the class of log-linear models, based on either a Poisson or (product) multinomial response distribution, is a flexible model class for inference and prediction. In this paper we consider the Bayesian analysis of both Poisson and multinomial log-linear models. It is often convenient to model multinomial or product multinomial data as observations of independent Poisson variables. For multinomial data, Lindley (1964) showed that this approach leads to valid Bayesian posterior inferences when the prior density for the Poisson cell means factorises in a particular way. We develop this result to provide a general framework for the analysis of multinomial or product multinomial data using a Poisson log-linear model. Valid finite population inferences are also available, which can be particularly important in modelling social data.We then focus particular attention on multivariate normal prior distributions for the log-linear model parameters.Here, an improper prior distribution for certain Poisson model parameters is required for valid multinomial analysis, and we derive conditions under which the resulting posterior distribution is proper.We also consider the construction of prior distributions across models, and for model parameters, when uncertainty exists about the appropriate form of the model. We present classes of Poisson and multinomial models, invariant under certain natural groups of permutations of the cells. We demonstrate that, if prior belief concerning the model parameters is also invariant, as is the case in a `reference' analysis, then choice of prior distribution is considerably restricted. The analysis of multivariate categorical data in the form of a contingency table is considered in detail. We illustrate the methods with two examples
Efficient construction of reversible jump Markov chain Monte Carlo proposal distributions - Discussion
No full textThe major implementational problem for reversible jump Markov chain Monte Carlo methods is that there is commonly no natural way to choose jump proposals since there is no Euclidean structure in the parameter space to guide our choice. We consider mechanisms for guiding the choice of proposal. The first group of methods is based on an analysis of acceptance probabilities for jumps. Essentially, these methods involve a Taylor series expansion of the acceptance probability around certain canonical jumps and turn out to have close connections to Langevin algorithms. The second group of methods generalizes the reversible jump algorithm by using the so–called saturated space approach. These allow the chain to retain some degree of memory so that, when proposing to move from a smaller to a larger model, information is borrowed from the last time that the reverse move was performed. The main motivation for this paper is that, in complex problems, the probability that the Markov chain moves between such spaces may be prohibitively small, as the probability mass can be very thinly spread across the space. Therefore, finding reasonable jump proposals becomes extremely important. We illustrate the procedure by using several examples of reversible jump Markov chain Monte Carlo applications including the analysis of autoregressive time series, graphical Gaussian modeling and mixture modelling.<br/
Ecological inference for 2 × 2 tables - Discussion
No full textA fundamental problem in many disciplines, including political science, sociology and epidemiology, is the examination of the association between two binary variables across a series of 2 × 2 tables, when only the margins are observed, and one of the margins is fixed. Two unobserved fractions are of interest, with only a single response per table, and it is this non-identifiability that is the inherent difficulty lying at the heart of ecological inference. Many methods have been suggested for ecological inference, often without a probabilistic model; we clarify the form of the sampling distribution and critique previous approaches within a formal statistical framework, thus allowing clarification and examination of the assumptions that are required under all approaches. A particularly difficult problem is choosing between models with and without contextual effects. Various Bayesian hierarchical modelling approaches are proposed to allow the formal inclusion of supplementary data, and/or prior information, without which ecological inference is unreliable. Careful choice of the prior within such models is required, however, since there may be considerable sensitivity to this choice, even when the model assumed is correct and there are no contextual effects. This sensitivity is shown to be a function of the number of areas and the distribution of the proportions in the fixed margin across areas. By explicitly providing a likelihood for each table, the combination of individual level survey data and aggregate level data is straightforward and we illustrate that survey data can be highly informative, particularly if these data are from a survey of the minority population within each area. This strategy is related to designs that are used in survey sampling and in epidemiology. An approximation to the suggested likelihood is discussed, and various computational approaches are described. Some extensions are outlined including the consideration of multiway tables, spatial dependence and area-specific (contextual) variables. Voter registration-race data from 64 counties in the US state of Louisiana are used to illustrate the methods. Copyright 2004 Royal Statistical Society.
Forecasting of cohort fertility under a hierarchical Bayesian approach
No full textFertility projections are a key determinant of population forecasts, which are widely used by government policy makers and planners. In keeping with the recent literature, we propose an intuitive and transparent hierarchical Bayesian model to forecast cohort fertility. Using Hamiltonian Monte Carlo methods and a data set from the human fertility database, we obtain fertility forecasts for 30 countries. We use scoring rules to assess the predictive accuracy of the forecasts quantitatively; these indicate that our model predicts with an accuracy comparable with that of the best-performing models in the current literature overall, with stronger performance for countries without a recent structural shift. Our findings support the position of hierarchical Bayesian modelling at the forefront of population forecasting methods.</p
Joint modelling of male and female mortality rates using adaptive P-splines
Full text linkRaw mortality data often exhibit irregular patterns due to randomness. Graduation refers to the act of smoothing crude mortality rates. In this paper, we propose a flexible and robust methodology for graduating mortality rates using adaptive P-splines. Since the observed data at high ages are often sparse and unreliable, we use an exponentially increasing penalty. We use mortality data of England and Wales and model male and female mortality rates jointly by means of penalties, achieving borrowing of information between the two sexes.</p
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