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Probability Based Independence Sampler for Bayesian Quantitative Learning in Graphical Log-Linear Marginal Models

Author

Listed:
  • Ioannis Ntzoufras

    (Department of Statistics, Athens University of Economics and Business)

  • Claudia Tarantola

    (Department of Economics and Management, University of Pavia)

  • Monia Lupparelli

    (Department of Statistical Sciences, University of Bologna)

Abstract

Bayesian methods for graphical log-linear marginal models has not been developed in the same extend as traditional frequentist approaches. In this work, we introduce a novel Bayesian approach for quantitative learning for such models. They belong to curved exponential families that are difficult to handle from a Bayesian perspective. Furthermore, the likelihood cannot be analytically expressed as a function of the marginal log-linear interactions, but only in terms of cell counts or probabilities. Posterior distributions cannot be directly obtained, and MCMC methods are needed. Finally, a well-defined model requires parameter values that lead to compatible marginal probabilities. Hence, any MCMC should account for this important restriction. We construct a fully automatic and efficient MCMC strategy for quantitative learning for graphical log-linear marginal models that handles these problems. While the prior is expressed in terms of the marginal log-linear interactions, we build an MCMC algorithm which employs a proposal on the probability parameter space. The corresponding proposal on the marginal log-linear interactions is obtained via parameter transformations. By this strategy, we achieve to move within the desired target space. At each step we directly work with well-defined probability distributions. Moreover, we can exploit a conditional conjugate setup to build an efficient proposal on probability parameters. The proposed methodology is illustrated by a simulation study and a real dataset.

Suggested Citation

  • Ioannis Ntzoufras & Claudia Tarantola & Monia Lupparelli, 2018. "Probability Based Independence Sampler for Bayesian Quantitative Learning in Graphical Log-Linear Marginal Models," DEM Working Papers Series 149, University of Pavia, Department of Economics and Management.
  • Handle: RePEc:pav:demwpp:demwp0149
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    File URL: http://dem-web.unipv.it/web/docs/dipeco/quad/ps/RePEc/pav/demwpp/DEMWP0149.pdf
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    References listed on IDEAS

    as
    1. Bartolucci, Francesco & Scaccia, Luisa & Farcomeni, Alessio, 2012. "Bayesian inference through encompassing priors and importance sampling for a class of marginal models for categorical data," Computational Statistics & Data Analysis, Elsevier, vol. 56(12), pages 4067-4080.
    2. Tamás Rudas & Wicher P. Bergsma, 2004. "On applications of marginal models for categorical data," Metron - International Journal of Statistics, Dipartimento di Statistica, Probabilità e Statistiche Applicate - University of Rome, vol. 0(1), pages 15-37.
    3. Thomas Richardson, 2003. "Markov Properties for Acyclic Directed Mixed Graphs," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 30(1), pages 145-157, March.
    4. Ntzoufras, Ioannis & Tarantola, Claudia, 2013. "Conjugate and conditional conjugate Bayesian analysis of discrete graphical models of marginal independence," Computational Statistics & Data Analysis, Elsevier, vol. 66(C), pages 161-177.
    5. Bahjat F. Qaqish & Anastasia Ivanova, 2006. "Multivariate logistic models," Biometrika, Biometrika Trust, vol. 93(4), pages 1011-1017, December.
    6. Mathias Drton & Thomas S. Richardson, 2008. "Binary models for marginal independence," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 70(2), pages 287-309, April.
    7. Robin J. Evans & Thomas S. Richardson, 2013. "Marginal log-linear parameters for graphical Markov models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 75(4), pages 743-768, September.
    8. Robin J. Evans, 2016. "Graphs for Margins of Bayesian Networks," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 43(3), pages 625-648, September.
    9. Monia Lupparelli & Giovanni M. Marchetti & Wicher P. Bergsma, 2009. "Parameterizations and Fitting of Bi‐directed Graph Models to Categorical Data," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 36(3), pages 559-576, September.
    10. Tamás Rudas & Wicher P. Bergsma & Renáta Németh, 2010. "Marginal log-linear parameterization of conditional independence models," Biometrika, Biometrika Trust, vol. 97(4), pages 1006-1012.
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    Cited by:

    1. Federica Nicolussi & Manuela Cazzaro, 2020. "Context-specific independencies in hierarchical multinomial marginal models," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 29(4), pages 767-786, December.

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    More about this item

    Keywords

    Graphical Models; Marginal Log-Linear Parameterisation; Markov Chain Monte Carlo Computation.;
    All these keywords.

    JEL classification:

    • E12 - Macroeconomics and Monetary Economics - - General Aggregative Models - - - Keynes; Keynesian; Post-Keynesian; Modern Monetary Theory
    • E21 - Macroeconomics and Monetary Economics - - Consumption, Saving, Production, Employment, and Investment - - - Consumption; Saving; Wealth
    • E22 - Macroeconomics and Monetary Economics - - Consumption, Saving, Production, Employment, and Investment - - - Investment; Capital; Intangible Capital; Capacity
    • E24 - Macroeconomics and Monetary Economics - - Consumption, Saving, Production, Employment, and Investment - - - Employment; Unemployment; Wages; Intergenerational Income Distribution; Aggregate Human Capital; Aggregate Labor Productivity
    • E31 - Macroeconomics and Monetary Economics - - Prices, Business Fluctuations, and Cycles - - - Price Level; Inflation; Deflation
    • C32 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes; State Space Models

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