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Nonconjugate priors and Metropolis-Hastings algorithms

# Nonconjugate priors and Metropolis-Hastings algorithms - 10...

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10 Nonconjugate priors and Metropolis-Hastings algorithms When conjugate or semiconjugate prior distributions are used, the posterior distribution can be approximated with the Monte Carlo method or the Gibbs sampler. In situations where a conjugate prior distribution is unavailable or undesirable, the full conditional distributions of the parameters do not have a standard form and the Gibbs sampler cannot be easily used. In this section we present the Metropolis-Hastings algorithm as a generic method of approx- imating the posterior distribution corresponding to any combination of prior distribution and sampling model. This section presents the algorithm in the context of two examples: The first involves Poisson regression, which is a type of generalized linear model. The second is a longitudinal regression model in which the observations are correlated over time. 10.1 Generalized linear models Example: Song sparrow reproductive success A sample from a population of 52 female song sparrows was studied over the course of a summer and their reproductive activities were recorded. In particular, the age and number of new offspring were recorded for each sparrow (Arcese et al, 1992). Figure 10.1 shows boxplots of the number of offspring versus age. The figure indicates that two-year-old birds in this population had the highest median reproductive success, with the number of offspring declining beyond two years of age. This is not surprising from a biological point of view: One-year-old birds are in their first mating season and are relatively inexperienced compared to two-year-old birds. As birds age beyond two years they experience a general decline in health and activity. Suppose we wish to fit a probability model to these data, perhaps to un- derstand the relationship between age and reproductive success, or to make population forecasts for this group of birds. Since the number of offspring for each bird is a non-negative integer { 0,1,2,. . . } , a simple probability model P.D. Hoff, A First Course in Bayesian Statistical Methods , Springer Texts in Statistics, DOI 10.1007/978-0-387-92407-6 10, c Springer Science+Business Media, LLC 2009

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172 10 Nonconjugate priors and Metropolis-Hastings algorithms 1 2 3 4 5 6 0 1 2 3 4 5 6 7 age offspring Fig. 10.1. Number of offspring versus age. for Y =number of offspring conditional on x =age would be a Poisson model, { Y | x } ∼ Poisson( θ x ). One possibility would be to estimate θ x separately for each age group. However, the number of birds of each age is small and so the estimates of θ x would be imprecise. To add stability to the estimation we will assume that the mean number of offspring is a smooth function of age. We will want to allow this function to be quadratic so that we can represent the increase in mean offspring while birds mature and the decline they experience thereafter. One possibility would be to express θ x as θ x = β 1 + β 2 x + β 3 x 2 .
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Nonconjugate priors and Metropolis-Hastings algorithms - 10...

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