Abstract
This chapter starts with a short survey of available software for the Bayesian analysis (Section 9.1), with the emphasis laid on high-level programming languages and free, open-source software issued under the GNU General Public License. Further, in Section 9.2, the WinBUGS environment for Bayesian analysis, applied to the analyses presented in this book, is described in more detail. In Section 9.3 an example of forecasting immigration flows using Bayesian approach, consisting of the estimation of the posterior distributions by means of the Gibbs algorithm, is demonstrated using the R environment, along with the Carlin–Chib (1995) model selection procedure (presented in Chapter 5). Finally, Section 9.4 offers general reflections on the advantages and limitations of particular software environments in practical applications.
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Contributed by Arkadiusz Wiśniowski
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For more details see http://www.gnu.org.
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Available at mathstat.helsinki.fi/openbugs/Manuals/Manual.html.
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For details, see the JAGS manual: www-fis.iarc.fr/~martyn/software/jags/jags_user_manual.pdf.
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A chain is said to be ‘well-mixing’ if it explores (almost) the whole parameter space. As opposed to a well-mixed chain, the poorly-mixed one stays in a small region for a long time.
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Description of the techniques is based on the OpenBUGS Manual (2009).
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‘Running mean’ or ‘running quantile’ denotes a series of means (or quantiles) calculated at each k-th iteration using the whole backward sample. It can be calculated recursively, e.g, a formula for a recursive mean is \(\bar x_{n + 1} = (n\bar x_n + x_{n + 1} )(n + 1)^{ - 1}\).
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Autocorrelation function (ACF) depicts correlations between sample values at different moments in time.
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Source: http://www.statbank.dk, accessed on 23 October 2009.
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Note that these are different flows than emigration introduced in Chapter 2 and denoted by \(M_{i - j(t)}\).
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Here, Gamma distribution is parameterised as \(p(x\left| {a,s} \right.) = s^{ - a} \Gamma (s)^{ - 1} x^{s - 1} \exp ( - x/s)\) with mean as and variance as 2.
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If A is a symmetric, positive-definite matrix, then the upper triangular factor of the Choleski decomposition is a matrix U, with strictly positive diagonal elements, such that U ’ U = A (after: MathWorld, mathworld.wolfram.com/CholeskyDecomposition.html, accessed on 10 December 2009).
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Cross-correlation measures the correlation of different series (here, parameters) at different moments in time.
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Technically, thinning can be introduced by means of the ‘modulo’ operation, and over-relaxed sampling for example by using a double loop and a function which returns a minimal value from a given set.
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Bijak, J. (2011). Bayesian Computing in Practice. In: Forecasting International Migration in Europe: A Bayesian View. The Springer Series on Demographic Methods and Population Analysis, vol 24. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-8897-0_9
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