Advertisement

Bayesian Models

  • Thomas OtterEmail author
Living reference work entry

Abstract

Bayesian models have become a mainstay in the tool set for marketing research in academia and industry practice. In this chapter, I discuss the advantages the Bayesian approach offers to researchers in marketing, the essential building blocks of a Bayesian model, Bayesian model comparison, and useful algorithmic approaches to fully Bayesian estimation. I show how to achieve feasible Bayesian inference to support marketing decisions under uncertainty using the Gibbs sampler, the Metropolis Hastings algorithm, and point to more recent developments – specifically the no-U-turn implementation of Hamiltonian Monte Carlo sampling available in Stan. The emphasis is on the development of an appreciation of Bayesian inference techniques supported by references to implementations in the open source software R, and not on the discussion of individual models. The goal is to encourage researchers to formulate new, more complete, and useful prior structures that can be updated with data for better marketing decision support.

Keywords

Marketing decision-making Bayesian inference Gibbs sampling Metropolis Hastings Hamiltonian Monte Carlo R bayesm Stan 

Notes

Acknowledgments

I would like to thank Anocha Aribarg, Albert Bemmaor, Joachim Büschken, Arash Laghaie, anonymous reviewers, the editors, and participants in my class on “Bayesian Modeling for Marketing” helpful comments and feedback. All remaining errors are obviously mine.

References

  1. Albert, J. H., & Chib, S. (1993). Bayesian analysis of binary and polychotomous response data. Journal of the American Statistical Association, 88(422), 669–679. http://www.jstor.org/stable/2290350
  2. Allenby, G. M., Arora, N., & Ginter, J. L. (1995). Incorporating prior knowledge into the analysis of conjoint studies. Journal of Marketing Research, 32(2), 152–162. http://www.jstor.org/stable/3152044
  3. Allenby, G. M., Arora, N., & Ginter, J. L. (1998). On the heterogeneity of demand. Journal of Marketing Research, 35(3), 384–389. http://www.jstor.org/stable/3152035
  4. Amemiya, T. (1985). Advanced econometrics. Cambridge, MA: Harvard University Press.Google Scholar
  5. Baron, R. M., & Kenny, D. A. (1986). The moderator–mediator variable distinction in social psychological research: Conceptual, strategic, and statistical considerations. Journal of Personality and Social Psychology, 51, 1173–1182.  https://doi.org/10.1037/0022-3514.51.6.1173.CrossRefGoogle Scholar
  6. Bernardo, J. M., & Smith, A. F. M. (2001). Bayesian theory. Measurement Science and Technology, 12(2), 221. http://stacks.iop.org/0957-0233/12/i=2/a=702.Google Scholar
  7. Besag, J. (1974). Spatial interaction and the statistical analysis of lattice systems. Journal of the Royal Statistical Society. Series B (Methodological), 36(2), 192–236. http://www.jstor.org/stable/2984812 Google Scholar
  8. Carpenter, B., Gelman, A., Hoffman, M., Lee, D., Goodrich, B., Betancourt, M., Brubaker, M., Guo, J., Li, P., & Riddell, A. (2017). Stan: A probabilistic programming language. Journal of Statistical Software, Articles, 76(1), 1–32.  https://doi.org/10.18637/jss.v076.i01. https://www.jstatsoft.org/v076/i01.CrossRefGoogle Scholar
  9. Chen, M.-H., Shao, Q.-M., & Ibrahim, J. G. (2000). Monte Carlo methods in Bayesian computation. New York: Springer. http://gateway.library.qut.edu.au/login?url=http://link.springer.com/openurl?genre=book&isbn=978-1-4612-1276-8.CrossRefGoogle Scholar
  10. Chib, S., & Carlin, B. P. (1999). On MCMC sampling in hierarchical longitudinal models. Statistics and Computing, 9(1), 17–26.  https://doi.org/10.1023/A:1008853808677.CrossRefGoogle Scholar
  11. Eddelbuettel, D. (2013). Seamless R and C+ + integration with Repp. New York: Springer.CrossRefGoogle Scholar
  12. Eddelbuettel, D., & François, R. (2011). Repp: Seamless R and C++ integration. Journal of Statistical Software, 40(8), 1–18.  https://doi.org/10.18637/jss.v040.i08. http://www.jstatsoft.org/v40/i08/
  13. Edwards, Y. D., & Allenby, G. M. (2003). Multivariate analysis of multiple response data. Journal of Marketing Research, 40(3), 321–334.  https://doi.org/10.1509/jmkr.40.3.321.19233.CrossRefGoogle Scholar
  14. Fasiolo, M. (2016). An introduction to mvnfast. R package version 0.1.6. https://CRAN.R-project.org/package=mvnfast
  15. Frühwirth-Schnatter, S., Tüchler, R., & Otter, T. (2004). Bayesian analysis of the heterogeneity model. Journal of Business & Economic Statistics, 22(1), 2–15.  https://doi.org/10.1198/073500103288619331.CrossRefGoogle Scholar
  16. Genz, A., Bretz, F., Miwa, T., Mi, X., Leisch, F., Scheipl, F., & Hothorn, T. (2018). mvtnorm: Multivariate normal and t distributions. https://CRAN.R-project.org/package=mvtnorm. R package version 1.0-8.
  17. Geweke, John. (1991). Efficient simulation from the multivariate normal and student-t distributions subject to linear constraints and the evaluation of constraint probabilities. In: E. M. Keramidas (Ed.), Computing Science and Statistics: Proceedings of the 23rd Symposium on the Interface, pp. 571–578.Google Scholar
  18. Gilks, W. R. (1996). Full conditional distributions. In S. (Sylvia) Richardson, D. J Spiegelhalter, & W. R. (Walter R.) Gilks (Eds.), Markov chain Monte Carlo in practice (pp. 75–88). London/Melbourne: Chapman & Hall.Google Scholar
  19. Hastie, T., Tibshirani, R., & Friedman, J. H. (2001). The elements of statistical learning: data mining, inference, and prediction. New York: Springer.CrossRefGoogle Scholar
  20. Hoerl, A. E., & Kennard, R. W. (1970). Ridge regression: Biased estimation for nonorthogonal problems. Technometrics, 12(1), 55–67.  https://doi.org/10.1080/00401706.1970.10488634.CrossRefGoogle Scholar
  21. Hoffman, M. D., & Gelman, A. (2014). The no-U-turn sampler: Adaptively setting path lengths in Hamiltonian Monte Carlo. Journal of Machine Learning Research, 15, 1593–1623. http://jmlr.org/papers/vl5/hoffmanl4a.html.Google Scholar
  22. Kass, R. E., & Raftery, A. E. (1995). Bayes factors. Journal of the American Statistical Association, 90(430), 773–795.  https://doi.org/10.1080/01621459.1995.10476572.CrossRefGoogle Scholar
  23. Lenk, P. J., & DeSarbo, W. S. (2000). Bayesian inference for finite mixtures of generalized linear models with random effects. Psychometrika, 65(1), 93–119.  https://doi.org/10.1007/BF02294188.CrossRefGoogle Scholar
  24. Lenk, P. J., DeSarbo, W. S., Green, P. E., & Young, M. R. (1996). Hierarchical Bayes conjoint analysis: Recovery of partworth heterogeneity from reduced experimental designs. Marketing Science, 15(2), 173–191.  https://doi.org/10.1287/mksc.15.2.173.CrossRefGoogle Scholar
  25. Long, J. S. (1997). Regression models for categorical and limited dependent variables. Thousand Oaks: Sage Publications. https://uk.sagepub.com/en-gb/eur/regression-models-for-categorical-and-limited-dependent-variables/book6071.Google Scholar
  26. McCulloch, R., & Rossi, P. (1994). An exact likelihood analysis of the multinomial probit model. Journal of Econometrics, 64(1–2), 207–240. https://EconPapers.repec.org/RePEc:eee:econom:v:64:y:1994:i:1-2:p:207-240.CrossRefGoogle Scholar
  27. Mersmann, O., Trautmann, H., Steuer, D., & Bornkamp, B. (2018). truncnorm: Truncated normal distribution. https://CRAN.R-project.org/package=truncnorm. R package version 1.0-8
  28. Montgomery, A. L., & Bradlow, E. T. (1999). Why analyst overconfidence about the functional form of demand models can lead to overpricing. Marketing Science, 18(4), 569–583. http://www.jstor.org/stable/193243
  29. Neal, R. M. (2011). MCMC using Hamiltonian dynamics. In S. Brooks, A. Gelman, G. L. Jones, & X-L. Meng (Eds.), Handbook of Markov chain Monte Carlo (Chap. 5). Chapman & Hall/CRC. http://arxiv.org/abs/1206.1901
  30. Orme, B. (2017). The CBC system for choice-based conjoint analysis. Technical Report. https://sawtoothsoftware.com/download/techpap/cbctech.pdf
  31. Otter, T., Tüchler, R., & Frühwirth-Schnatter, S. (2004). Capturing consumer heterogeneity in metric conjoint analysis using Bayesian mixture models. International Journal of Research in Marketing, 21(3), 285–297.  https://doi.org/10.1016/j.ijresmar.2003.11.002. http://www.sciencedirect.com/science/article/pii/S0167811604000308 CrossRefGoogle Scholar
  32. Otter, T., Gilbride, T. J., & Allenby, G. M. (2011). Testing models of strategic behavior characterized by conditional likelihoods. Marketing Science, 30(4), 686–701. http://www.jstor.org/stable/23012019
  33. Otter, T., Pachali, M. J., Mayer, S., & Landwehr, J. R. (2018). Causal inference using mediation analysis or instrumental variables – Full mediation in the absence of conditional independence. Marketing ZFP, 40(2), 41–57.  https://doi.org/10.15358/0344-1369-2018-2-41.CrossRefGoogle Scholar
  34. Pachali, M. J., Kurz, P., & Otter, T. (2018). How to generalize from a hierarchical model? Technical Report. https://ssrn.com/abstract=3018670
  35. Pearl, J. (2009). Causality: Models, reasoning and inference (2nd ed.). New York: Cambridge University Press.CrossRefGoogle Scholar
  36. Plummer, M., Best, N., Cowles, K., & Vines, K. (2006). Coda: Convergence diagnosis and output analysis for MCMC. R News, 6(1), 7–11. https://journal.r-project.org/archive/.Google Scholar
  37. Ritter, C., & Tanner, M. A. (1992). Facilitating the Gibbs sampler: The Gibbs stopper and the Griddy-Gibbs sampler. Journal of the American Statistical Association, 87(419), 861–868.  https://doi.org/10.1080/01621459.1992.10475289.CrossRefGoogle Scholar
  38. Robert, C. P. (1994). The Bayesian choice: a decision-theoretic motivation. New York: Springer.CrossRefGoogle Scholar
  39. Roberts, G. O. (1996). Markov chain concepts related to sampling algorithms. In S. (Sylvia) Richardson, D. J. Spiegelhalter, & W. R. (Walter R.) Gilks (Eds.), Markov chain Monte Carlo in practice (pp. 45–58). London/Melbourne: Chapman & Hall.Google Scholar
  40. Rossi, P. E., McCulloch, R. E., & Allenby, G. M. (1996). The value of purchase history data in target marketing. Marketing Science, 15(4), 321–340.  https://doi.org/10.1287/mksc.l5.4.321.CrossRefGoogle Scholar
  41. Rossi, P. E., Allenby, G. M., & McCulloch, R. E. (2005). Bayesian statistics and marketing. Chichester: Wiley.CrossRefGoogle Scholar
  42. Wachtel, S., & Otter, T. (2013). Successive sample selection and its relevance for management decisions. Marketing Science, 32(1), 170–185.  https://doi.org/10.1287/mksc.1120.0754.CrossRefGoogle Scholar
  43. Zellner, A. (1971). An introduction to Bayesian inference in econometrics. New York: Wiley.Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Goethe University Frankfurt am MainFrankfurt am MainGermany

Personalised recommendations