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Bayesian Analysis

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Advanced Methods for Modeling Markets

Part of the book series: International Series in Quantitative Marketing ((ISQM))

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Abstract

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Notes

  1. 1.

    It may seem extreme to cite such an ancient reference. However, Rev. Thomas Bayes’ essay, as presented by Richard Price to the Royal Society, is a remarkably sophisticated and modern treatise about probability and inference if one updates its eighteenth century notation to twenty-first century standards. It no doubt languished in the archives for centuries because it was that far ahead of the science of the time.

  2. 2.

    A surprising result is that our old friend, the sample mean, is not admissible under squared error loss in three or more dimensions (Stein 1956 and James and Stein 1961)!

  3. 3.

    Suppose that y = α + βx + ε with E(ε) = 0, E(x ε) = 0, and β ≠ 0. Then E(y| x) = α + βx. Further suppose that E(y) = E(x) = 0. Then E(y) = E[E(y| x)] = α + βE(x). Since E(y) = E(x) = 0, then α = 0.

  4. 4.

    The MLE is also the least-squares estimate that minimizes sums-of-squares errors: \( \sum_{i=1}^n{\left({y}_i-\beta {x}_i\right)}^2. \) Compare with Sect. 6.4.1, Vol. I and Sect. 1.4.1 in this volume.

  5. 5.

    But amazingly good looking (for statisticians).

  6. 6.

    Exponential families include almost every distribution we can easily write down, except the uniform and t-distribution.

  7. 7.

    Frequentists often use the subjunctive mood when describing inference or hypothesis testing: “The null hypothesis would be rejected in 5% of all random samples if it were true.” Bayesian use the indicative mood: “The posterior probability of the null hypothesis is 0.05 given our data and model.”

  8. 8.

    If we modify the prior distributions slightly, they are conjugate and have a closed form. The modification conditions the slope on the error variance: \( \beta\,{\mid}\,{\sigma}^2\,{\sim}\,N\left({b}_0,{\sigma}^2{v}_0^2\right) \) and \( {\sigma}^2\,{\sim}\,IG\left(\frac{r_0}{2},{s}_0/2\right) \). This results in a “scale free” model: you can multiply y and x by a constant without changing the prior distributions. Instead of pursuing the algebra for this specification, we think it is a better use of time to introduce numerical methods.

  9. 9.

    It is strange nomenclature. Metropolis et al. (1953) generated random variables from the Gibbs distribution, which describes the distribution of energy states for a system of atoms. Because they were sampling from a Gibbs distribution, this special case is called “Gibbs sampling” even when it is not applied to a Gibbs distribution.

  10. 10.

    In multivariate regression models, β would be the vector of regression coefficients.

  11. 11.

    In fairness to Newton and Raftery (1994), their method is not the main focus of their paper.

  12. 12.

    Some Bayesians refer to the distribution on β i |μ β , V β as the “prior” and the distributions μ|μ 0, V 0 and V|r, R as “hyperpriors.” Others view β i |μ β , V β as part of the random-coefficients “model.” While this occasionally causes some confusion among Bayesians, the difference is unimportant in practice, as both the probability model and the priors should be carefully considered and should represent the analyst’s subjective beliefs. In this chapter we have avoided making a strong distinction between models and prior.

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Authors and Affiliations

  1. LeBow College of Business, Drexel University, 3220 Market Street, Philadelphia, PA, 19104, USA

    Elea McDonnell Feit

  2. Ross School of Business, University of Michigan, 701 Tappan St., Ann Arbor, MI, 48109, USA

    Fred M. Feinberg & Peter J. Lenk

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  1. Elea McDonnell Feit
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Editors and Affiliations

  1. Department of Marketing, University of Groningen, Groningen, The Netherlands

    Peter S. H. Leeflang

  2. Department of Marketing, University of Groningen, Groningen, The Netherlands

    Jaap E. Wieringa

  3. Department of Marketing, University of Groningen, Groningen, The Netherlands

    Tammo H.A Bijmolt

  4. Department of Marketing, D’Amore-McKim School of Business, Northeastern University, Boston, USA

    Koen H. Pauwels

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Feit, E.M., Feinberg, F.M., Lenk, P.J. (2017). Bayesian Analysis. In: Leeflang, P., Wieringa, J., Bijmolt, T., Pauwels, K. (eds) Advanced Methods for Modeling Markets. International Series in Quantitative Marketing. Springer, Cham. https://doi.org/10.1007/978-3-319-53469-5_16

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