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A Bayesian Vector Multidimensional Scaling Procedure Incorporating Dimension Reparameterization with Variable Selection

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Abstract

We propose a two-way Bayesian vector spatial procedure incorporating dimension reparameterization with a variable selection option to determine the dimensionality and simultaneously identify the significant covariates that help interpret the derived dimensions in the joint space map. We discuss how we solve identifiability problems in a Bayesian context that are associated with the two-way vector spatial model, and demonstrate through a simulation study how our proposed model outperforms a popular benchmark model. In addition, an empirical application dealing with consumers’ ratings of large sport utility vehicles is presented to illustrate the proposed methodology. We are able to obtain interpretable and managerially insightful results from our proposed model with variable selection in comparison with the benchmark model.

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References

  • Addelman, S. (1961). Irregular fractions of the \(2^{{\rm n}}\) factorial experiments. Technometrics, 3, 479–496.

    Google Scholar 

  • Barbieri, M. M., & Berger, J. (2004). Optimal predictive model selection. Annals of Statistics, 32, 870–897.

    Article  Google Scholar 

  • Benzecri, J. P. (1992). Correspondence analysis handbook. New York: Marcel Dekker.

    Google Scholar 

  • Bolton, G. E., Fong, D. K. H., & Mosquin, P. (2003). Bayes factors with an application to experimental economics. Experimental Economics, 6, 311–325.

    Article  Google Scholar 

  • Borg, I., & Groenen, P. J. F. (2005). Modern multidimensional scaling: Theory and applications (2nd ed.). New York: Springer.

    Google Scholar 

  • Brown, P. J., Vannucci, M., & Fearn, T. (1998). Multivariate Bayesian variable selection and prediction. Journal of the Royal Statistical Society Series B, 60, 627–641.

    Article  Google Scholar 

  • Buja, N., & Eyuboglu, N. (1992). Remarks on parallel analysis. Multivariate Behavioral Research, 27(4), 509–540.

    Article  Google Scholar 

  • Carroll, J. D. (1980). Models and methods for multidimensional analysis of preferential choice (or other dominance) data. In E. D. Lantermann & H. Feger (Eds.), Similarity and choice (pp. 234–289). Vienna: Hans Huber Publishers.

    Google Scholar 

  • Carroll, J. D., & Arabie, P. (1980). Multidimensional scaling. Annual Review of Psychology, 31, 607–649.

    Article  PubMed  Google Scholar 

  • Carroll, J. D., Pruzanksy, S., & Kruskal, J. B. (1980). CANDELINC: A general approach to multidimensional analysis of many-way arrays with linear constraints on parameters. Psychometrika, 45(1), 3–24.

    Article  Google Scholar 

  • Cox, T. F., & Cox, M. A. A. (2001). Multidimensional scaling (2nd ed.). Boca Raton, FL: Chapman and Hall/CRC.

    Google Scholar 

  • Dawid, A. (1981). Some matrix-variate distribution theory: Notational considerations and a Bayesian application. Biometrika, 68(1), 265–274.

    Article  Google Scholar 

  • DeSarbo, W. S. (1982). GENNCLUS: New models for general nonhierarchical clustering analysis. Psychometrika, 47(4), 449–475.

    Article  Google Scholar 

  • DeSarbo, W. S., & Carroll, J. D. (1985). Three-way metric unfolding via alternating weighted least squares. Psychometrika, 50(3), 275–300.

    Article  Google Scholar 

  • DeSarbo, W. S., & Cho, J. (1989). A stochastic multidimensional scaling vector threshold model for the spatial representation of pick any/N data. Psychometrika, 54(1), 105–121.

    Article  Google Scholar 

  • DeSarbo, W. S., Fong, D. K. H., Liechty, J., & Saxton, K. (2004). A hierarchical Bayesian procedure for two-mode cluster analysis. Psychometrika, 69, 547–572.

    Article  Google Scholar 

  • DeSarbo, W. S., Grewal, R., & Scott, C. J. (2008). A clusterwise bilinear multidimensional scaling methodology for simultaneous segmentation and positioning analysis. Journal of Marketing Research, 45(2), 280–292.

    Article  Google Scholar 

  • DeSarbo, W. S., Howard, D. J., & Jedidi, K. (1991). MULTICLUS: A new method for simultaneously performing multidimensional scaling and cluster analysis. Psychometrika, 56(1), 121–136.

    Article  Google Scholar 

  • DeSarbo, W. S., & Jedidi, K. (1995). The spatial representation of heterogeneous consideration sets. Marketing Science, 14(3), 326–342.

    Article  Google Scholar 

  • DeSarbo, W. S., & Kim, S. (2013). A review of the major multidimensional scaling models for the analysis of preference/dominance data in marketing. In L. Moutinho & K.-H. Huarng (Eds.), Quantitative Modeling in Marketing and Management (pp. 3–27). London: World Scientific Press.

    Google Scholar 

  • DeSarbo, W. S., Kim, Y., & Fong, D. K. H. (1999). A Bayesian multidimensional scaling procedure for the spatial analysis of revealed choice data. Journal of Econometrics, 89, 79–108.

    Article  Google Scholar 

  • DeSarbo, W. S., Oliver, R. L., & DeSoete, G. (1986). A probabilistic multidimensional scaling vector model. Applied Psychological Measurement, 10(1), 79–98.

    Article  Google Scholar 

  • DeSarbo, W. S., Park, J., & Rao, V. (2011). Deriving joint space positioning maps from consumer preference ratings. Marketing Letters, 22(1), 1–14.

    Article  Google Scholar 

  • DeSarbo, W. S., & Rao, V. R. (1986). A constrained unfolding methodology for product positioning. Marketing Science, 5, 1–19.

    Article  Google Scholar 

  • Fong, D. K. H. (2010). Bayesian multidimensional scaling and its applications in marketing research. In Ming-Hui Chen, Dipak K. Dey, Peter Mueller, Dongchu Sun, & Keying Ye (Eds.), Frontier of Statistical Decision Making and Bayesian Analysis (pp. 410–417). Berlin: Springer.

    Google Scholar 

  • Fong, D. K. H., DeSarbo, W. S., Park, J., & Scott, C. J. (2010). A Bayesian vector multidimensional scaling procedure for the analysis of ordered preference data. Journal of the American Statistical Association, 105(490), 482–492.

    Article  Google Scholar 

  • Fong, D. K. H., Ebbes, P., & DeSarbo, W. S. (2012). A heterogeneous Bayesian regression model for cross sectional data involving a single observation per response unit. Psychometrika, 77(2), 293–314.

    Article  Google Scholar 

  • George, E. I., & McCulloch, R. E. (1993). Variable selection via gibbs sampling. Journal of American Statistical Association, 88, 881–889.

    Article  Google Scholar 

  • George, E. I., & McCulloch, R. E. (1997). Approaches for Bayesian variable selection. Statistica Sinica, 7, 339–373.

    Google Scholar 

  • Gifi, A. (1990). Nonlinear multivariate analysis. Chichester, England: Wiley.

    Google Scholar 

  • Golub, G. H., & Van Loan, C. F. (1996). Matrix computations (3rd ed.). Baltimore, MD: Johns Hopkins University Press.

    Google Scholar 

  • Gormley, I. C., & Murphy, T. B. (2006). A latent space model for rank data. In Statistical network analysis: Models, issues and new directions. Lecture notes in computer science. New York: Springer. Available as technical report at http://www.tcd.ie/Statistics/postgraduate/0602.pdf.

  • Gupta, A. K., & Nagar, D. K. (2000). Matrix variate distributions. Monographs and surveys in pure and applied mathematics (Vol. 104). London: Chapman & Hall/CRC.

    Google Scholar 

  • Gustafson, P. (2005). On model expansion, model contraction, identifiability and prior information: Two illustrative scenarios involving mismeasured variables. Statistical Science, 20, 111–140.

    Article  Google Scholar 

  • Harshman, R. A., & Lundy, M. E. (1984). Data preprocessing and the extended PARAFAC model. In H. G. Law, C. W. Snyder Jr, J. Hattie, & R. P. McDonald (Eds.), Research methods for multimode data analysis (pp. 216–284). New York: Praeger.

    Google Scholar 

  • Jedidi, K., & DeSarbo, W. S. (1991). A stochastic multidimensional scaling procedure for the spatial representation of three-mode, three-way pick any/J data. Psychometrika, 56(3), 471–494.

    Article  Google Scholar 

  • Jolliffe, I. T., Trendafilov, N. T., & Uddin, M. (2003). A modified principal component technique based on the LASSO. Journal of Computational and Graphical Statistics, 12(3), 531–547.

    Article  Google Scholar 

  • Kass, R. E., & Raftery, A. E. (1995). Bayes factors. Journal of the American Statistical Association, 90, 773–795.

    Article  Google Scholar 

  • Kruskal, J. B. (1964). Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis. Psychometrika, 29, 1–27.

    Article  Google Scholar 

  • Lee, M. D. (2008). Three case studies in the Bayesian analysis of cognitive models. Psychonomic Bulletin & Review, 15, 1–15.

    Article  Google Scholar 

  • Oh, M.-S., & Raftery, A. E. (2001). Bayesian multidimensional scaling and choice of dimension. Journal of the American Statistical Association, 96, 1031–1044.

    Article  Google Scholar 

  • O’Hara, R. B. O., & Sillanpaa, M. J. (2009). A review of Bayesian variable selection methods: What, how and which. Bayesian Analysis, 4(1), 85–118.

    Article  Google Scholar 

  • Park, J., DeSarbo, W. S., & Liechty, J. (2008). A hierarchical Bayesian multidimensional scaling methodology for accommodating both structural and preference heterogeneity. Psychometrika, 73(3), 451–472.

    Article  Google Scholar 

  • Raftery, A. E., Newton, M. A., Satagopan, J. M., & Krivitsky, P. N. (2007). Estimating the integrated likelihood via posterior simulation using the harmonic mean identity. In J. M. Bernardo, M. J. Bayarri, J. O. Berger, A. P. Dawid, D. Heckerman, A. F. M. Smith, & M. West (Eds.), Bayesian statistics 8 (pp. 1–45). Oxford: Oxford University Press.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • Schonemann, P. H. (1970). On metric multidimensional unfolding. Psychometrika, 35(3), 349–366.

    Article  Google Scholar 

  • Scott, C. J., & DeSarbo, W. S. (2011). A new constrained stochastic multidimensional scaling vector model: An application to the perceived importance of leadership attributes. Journal of Modeling in Management, 6(1), 7–32.

    Article  Google Scholar 

  • Shepard, R. N. (1962). The analysis of proximities: Multidimensional scaling with an unknown distance function. Psychometrika, 27(125–140), 219–246.

    Article  Google Scholar 

  • Shepard, R. N. (1980). Multidimensional scaling, tree-fitting, and clustering. Science, 210, 390–398.

    Article  PubMed  Google Scholar 

  • Shin, J. S., Fong, D. K. H., & Kim, K. J. (1998). Complexity reduction of a house of quality chart using correspondence analysis. Quality Management Journal, 5, 46–58.

    Google Scholar 

  • Slater, P. (1960). The analysis of personal preferences. The British Journal of Statistical Psychology, 13, 119–135.

    Article  Google Scholar 

  • Spiegelhalter, D. J., Best, N. G., Carlin, B. P., & van der Linde, A. (2002). Bayesian measures of model complexity and fit. Journal of the Royal Statistical Society, Series B, 64, 583–639.

    Article  Google Scholar 

  • Takane, Y. (2013). Constrained principal component analysis. New York, NY: Chapman & Hall Inc.

    Google Scholar 

  • Takane, Y., Young, F., & Leeuw, J. (1977). Nonmetric individual differences multidimensional scaling: An alternating least squares method with optimal scaling features. Psychometrika, 42(1), 7–67.

    Article  Google Scholar 

  • Ter Braak, C. J. F. (1986). Canonical correspondence analysis: A new eigenvector technique for multivariate direct gradient analysis. Ecology, 67(5), 1167–1179.

    Article  Google Scholar 

  • Tucker, L. R. (1960). Intra-individual and inter-individual multidimensionality. In H. Gullikson & S. Messick (Eds.), Psychological Scaling: Theory and Applications. New York, NY: Holt, Rinehart, & Winston.

    Google Scholar 

Download references

Acknowledgments

The authors wish to thank the editor, associate editor, and three anonymous referees for their constructive comments. This research was funded in part by the Smeal College of Business.

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Authors and Affiliations

  1. Department of Marketing, Pennsylvania State University, University Park, PA, 16802, USA

    Duncan K. H. Fong & Wayne S. DeSarbo

  2. Department of Statistics, Pennsylvania State University, University Park, PA, 16802, USA

    Zhe Chen & Zhuying Xu

Authors
  1. Duncan K. H. Fong
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  2. Wayne S. DeSarbo
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  3. Zhe Chen
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  4. Zhuying Xu
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Corresponding author

Correspondence to Duncan K. H. Fong.

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Zhe Chen is currently working at Google Inc.

Appendices

Appendix 1: Full Conditional Distributions

(i) Let \({\varvec{Z}}\) be the data matrix. Since

$$\begin{aligned}&{\pi (\sigma ^{-2}|\hbox {all others})} \\\propto & {} {(\sigma ^{-2})^{NJ/2}\hbox {etr}\left\{ -\frac{\sigma ^{-2}}{2}({\varvec{Z}}-{\varvec{A}}^{{\prime }}{\varvec{B}})({\varvec{Z}}- {\varvec{A}}^{{\prime }}{\varvec{B}})^{{\prime }}\right\} (\sigma ^{-2})^{m_1 -1}\hbox {exp}\{-m_2 \sigma ^{-2}\}} \\= & {} {(\sigma ^{-2})^{\frac{NJ}{2}+m_1 -1}\hbox {exp}\left\{ {-\sigma ^{-2}\left[ {m_2 +\frac{1}{2}\hbox {tr}\left[ {\left( {{\varvec{Z}}-{\varvec{A}}^{{{\prime }}}{\varvec{B}}} \right) \left( {{\varvec{Z}}-{\varvec{A}}^{{{\prime }}}{\varvec{B}}} \right) ^{{{\prime }}}} \right] } \right] } \right\} ,} \\ \end{aligned}$$

where etr refers to an exponential function of the trace of (a matrix), the full conditional distribution of \(\sigma ^{-2}\) is \(\hbox {Ga}\left( {\hbox {m}_1^*,m_2^*} \right) \), where \(\hbox {m}_1^*=\left( {\frac{NJ}{2}+m_1 } \right) \hbox { and} \quad m_2^*=\left( {m_2 +\frac{1}{2}\hbox {tr}[({\varvec{Z}}-{\varvec{A}}^{{\prime }}{\varvec{B}}) ({\varvec{Z}}-{\varvec{A}}^{{\prime }}{\varvec{B}})^{{\prime }}]} \right) \).

(ii) Let \({\varvec{A}}_0 =\mathbf{1}^{{{\prime }}}{\otimes } {\varvec{a}}_0 \). Since

$$\begin{aligned}&{\pi ({\varvec{A}}|\hbox {all others})} \\\propto & {} {\hbox {etr}\left\{ {-\frac{1}{2}\left[ {\sigma ^{-2}\left( {{\varvec{Z}}-{\varvec{A}}^{{{\prime }}}{\varvec{B}}} \right) \left( {{\varvec{Z}}-{\varvec{A}}^{{{\prime }}}{\varvec{B}}} \right) ^{{{\prime }}}+({\varvec{A}}-{\varvec{A}}_0 )^{{\prime }} ({\varvec{A}}-{\varvec{A}}_0 )/c} \right] } \right\} } \\\propto & {} {\hbox {etr}\left\{ {-\frac{1}{2}\left[ {{\varvec{A}}^{{\prime }}(\sigma ^{-2}\mathbf{BB}^{{\prime }}+I_\mathrm{T} /c){\varvec{A}}-2{\varvec{A}}^{{\prime }}(\sigma ^{-2}\mathbf{B} {\varvec{Z}}^{{\prime }}+{\varvec{A}}_0 /c)} \right] } \right\} ,} \end{aligned}$$

the full conditional distribution of \({\varvec{A}}\) is \(\hbox {MN}\left( {\bar{{\varvec{A}}} ,{\varvec{I}}_N ,{\varvec{A}}_l } \right) \), where \({\varvec{A}}_l =(\sigma ^{-2}\mathbf{BB}^{{\prime }}+{\varvec{I}}_\mathrm{T} /c)^{-1}\hbox { and } \bar{{\varvec{A}}} ={\varvec{A}}_l (\sigma ^{-2}\mathbf{B}{\varvec{Z}}^{{\prime }}+{\varvec{A}}_0 /c)\).

(iii) Since

$$\begin{aligned}&{\pi ( {\varvec{a}}_0 |\hbox {all others})} \\\propto & {} {\hbox {exp}\left\{ {-\frac{1}{2} \sum \limits _{i=1}^N \left[ {( {\varvec{a}}_i - {\varvec{a}}_0 )^{{\prime }}( {\varvec{a}}_i - {\varvec{a}}_0 )/c} \right] -\frac{1}{2} {\varvec{a}}_0 ^{{\prime }}{\varvec{G}}_a^{-1} {\varvec{a}}_0 } \right\} } \\\propto & {} {\hbox {exp}\left\{ {-\frac{1}{2}\left[ { {\varvec{a}}_0 ^{{\prime }}\left( {\frac{N}{c}\mathbf{I}_\mathrm{T} +{\varvec{G}}_a^{-1} } \right) {\varvec{a}}_0 -2 {\varvec{a}}_0 ^{{\prime }}\left( { \sum \limits _{i=1}^N \,{\varvec{a}}_i } \right) /c} \right] } \right\} ,} \end{aligned}$$

the full conditional distribution of \({\varvec{a}}_0\) is \(N\left( {\bar{{\varvec{a}}} ,{\varvec{G}}_{an} } \right) \), where \({\varvec{G}}_{an} =\left( {\frac{N}{c}\mathbf{I}_\mathrm{T} +{\varvec{G}}_a^{-1} } \right) ^{-1}\hbox { and } \bar{{\varvec{a}}} ={\varvec{G}}_{an} \left( {\sum _{i=1}^N \,{\varvec{a}}_i } \right) /c\).

(iv) Let \({\varvec{B}}_0 =\mathbf{1}^{{{\prime }}}{\otimes } {\varvec{b}}_0 \). Since

$$\begin{aligned}&{\pi (\mathbf{B}|\hbox {all others})} \\\propto & {} {\hbox {etr}\left\{ {-\frac{1}{2}\left[ {\sigma ^{-2}\left( {{\varvec{Z}}-{\varvec{A}}^{{{\prime }}}{\varvec{B}}} \right) \left( {{\varvec{Z}}-{\varvec{A}}^{{{\prime }}}{\varvec{B}}} \right) ^{{{\prime }}}+(\mathbf{B}-\mathbf{B}_0 -{\varvec{\Theta }}{\varvec{X}})^{{\prime }}{\varvec{\Sigma }}^{-1}(\mathbf{B}-\mathbf{B}_0 -{\varvec{\Theta }}{\varvec{X}})} \right] } \right\} } \\\propto & {} {\hbox {etr}\left\{ {-\frac{1}{2} \left[ {{\varvec{B}}^{{\prime }}(\sigma ^{-2}{{\varvec{A}A}}^{{\prime }}+{\varvec{\Sigma }}^{-1}){\varvec{B}}- 2{\varvec{B}}^{{\prime }}(\sigma ^{-2}{{\varvec{A}Z}}+{\varvec{\Sigma }}^{-1}({\varvec{B}}_0 +{\varvec{\Theta }}{\varvec{X}}))} \right] } \right\} ,} \end{aligned}$$

the full conditional distribution of \(\mathbf{B}\) is \(\hbox {MN}\left( {\bar{{\varvec{B}}} ,{\varvec{I}}_J ,{\varvec{B}}_l } \right) \), where \({\varvec{B}}_l =(\sigma ^{-2}{{\varvec{A}A}}^{{\prime }}+{\varvec{\Sigma }}^{-1})^{-1}\hbox { and } \bar{{\varvec{B}}} ={\varvec{B}}_l (\sigma ^{-2}{{\varvec{A}Z}}+{\varvec{\Sigma }}^{-1}({\varvec{B}}_0 +{\varvec{\Theta }}{\varvec{X}}))\).

(v) Since

$$\begin{aligned}&{\pi ( {\varvec{b}}_0 |\hbox {all others})} \\\propto & {} {\hbox {exp}\left\{ {-\frac{1}{2} \sum \limits _{j=1}^J \,\left[ {( {\varvec{b}}_j - {\varvec{b}}_0 -{\varvec{\Theta }}{\varvec{X}}_j )^{{\prime }}{\varvec{\Sigma }}^{-1}( {\varvec{b}}_j - {\varvec{b}}_0 -{\varvec{\Theta }}{\varvec{X}}_j )} \right] -\frac{1}{2} {\varvec{b}}_0 ^{{\prime }}{\varvec{G}}_b^{-1} {\varvec{b}}_0 } \right\} } \\\propto & {} {\hbox {exp}\left\{ {-\frac{1}{2}\left[ { {\varvec{b}}_0 ^{{\prime }}\left( {J{\varvec{\Sigma }}^{-1}+{\varvec{G}}_b^{-1} } \right) {\varvec{b}}_0 -2 {\varvec{b}}_0 ^{{\prime }}{\varvec{\Sigma }}^{-1}\sum \limits _{j=1}^J \,\left( { {\varvec{b}}_j -{\varvec{\Theta }}{\varvec{X}}_j } \right) } \right] } \right\} ,} \end{aligned}$$

the full conditional distribution of \({\varvec{b}}_0\) is \(N\left( {\bar{{\varvec{b}}} ,{\varvec{G}}_{bn} } \right) \), where \({\varvec{G}}_{bn} =\left( {J{\varvec{\Sigma }}^{-1}+{\varvec{G}}_b^{-1} } \right) ^{-1}\hbox { and } \bar{{\varvec{b}}} ={\varvec{G}}_{bn} {\varvec{\Sigma }}^{-1} \sum _{j=1}^J \,\left( { {\varvec{b}}_j -{\varvec{\Theta }}{\varvec{X}}_j } \right) \).

(vi) Since

$$\begin{aligned}&\pi \left( {{\varvec{\Sigma }}^{-1}\hbox {|all others}} \right) \\&\propto |{\varvec{\Sigma }}^{-1}|^{\frac{J}{2}}\hbox {etr}\left\{ {-\frac{1}{2}(\mathbf{B}-\mathbf{B}_0 -{\varvec{\Theta }}_{\left( \gamma \right) } {\varvec{X}}_{\left( \gamma \right) } )^{{\prime }}{\varvec{\Sigma }}^{-1}(\mathbf{B}-\mathbf{B}_0 -{\varvec{\Theta }}_{\left( \gamma \right) } {\varvec{X}}_{\left( \gamma \right) } )} \right\} \\&\quad \times |{\varvec{\Sigma }}^{-1}|^{\frac{v-\mathrm{T}-1}{2}}\hbox {etr}\left\{ {-\frac{V^{-1}}{2}{\varvec{\Sigma }}^{-1}} \right\} |{\varvec{\Sigma }}^{-1}|^{\frac{\sum _k \,\gamma _k }{2}}\hbox {etr}\left\{ {-\frac{1}{2}{\varvec{\Theta }}_{(\gamma )} {\varvec{H}}_{(\gamma )}^{-1} {\varvec{\Theta }}_{(\gamma )}^{\prime } {\varvec{\Sigma }}^{-1}} \right\} \\&\propto |{\varvec{\Sigma }}^{-1}|^{\frac{J+\sum _k \,\gamma _k +\nu -\mathrm{T}-1}{2}} \\&\quad \times \, \hbox {etr}\left\{ {\frac{-1}{2}{\varvec{\Sigma }}^{-1}\left[ {\left( {\mathbf{B}-\mathbf{B}_0 -{\varvec{\Theta }}_{\left( \gamma \right) } {\varvec{X}}_{\left( \gamma \right) } } \right) \left( {\mathbf{B}-\mathbf{B}_0 -{\varvec{\Theta }}_{\left( \gamma \right) } {\varvec{X}}_{\left( \gamma \right) } } \right) ^{{{\prime }}}\!+\!V^{-1}{\varvec{I}}_\mathrm{T} +{\varvec{\Theta }}_{\left( \gamma \right) } {\varvec{H}}_{\left( \gamma \right) }^{-1} {\varvec{\Theta }}_{\left( \gamma \right) }^{\prime } } \right] } \right\} , \end{aligned}$$

the full conditional distribution of \({\varvec{\Sigma }}^{-1}\) is \(W(J+\sum _k \,\gamma _k +\nu ,{\varvec{V}}_n )\), where \({\varvec{V}}_n =\left[ \left( \mathbf{B}-\mathbf{B}_0 -{\varvec{\Theta }}_{\left( \gamma \right) }\right. \right. \) \(\left. \left. {\varvec{X}}_{\left( \gamma \right) } \right) \left( {\mathbf{B}-\mathbf{B}_0 -{\varvec{\Theta }}_{\left( \gamma \right) } {\varvec{X}}_{\left( \gamma \right) } } \right) ^{{{\prime }}}+V^{-1}{\varvec{I}}_\mathrm{T} +{\varvec{\Theta }}_{\left( \gamma \right) } {\varvec{H}}_{\left( \gamma \right) }^{-1} {\varvec{\Theta }}_{\left( \gamma \right) }^{\prime } \right] ^{-1}\).

(vii) Since

$$\begin{aligned}&{\pi (w|\hbox {all others})} \\\propto & {} {w^{p-1}(1-w)^{q-1} \prod \limits _{k=1}^K \,(w^{\gamma _k }(1-w)^{1-\gamma _k })} \\\propto & {} {w^{p+\sum _{k=1}^K \,\gamma _k -1}(1-w)^{q+K-\left( {\sum _{k=1}^K \,\gamma _k } \right) -1},} \end{aligned}$$

the full conditional distribution of \(w\) is \(\hbox {Beta}(p+\sum _{k=1}^K \,\gamma _k ,q+K-\sum _{k=1}^K \,\gamma _k )\).

(viii) Since

$$\begin{aligned}&{\pi ({\varvec{\Theta }}_{\left( \gamma \right) } |\hbox {all others})} \\\propto & {} {\hbox {etr}\left\{ {-\frac{1}{2}\left[ {(\mathbf{B}-\mathbf{B}_0 -{\varvec{\Theta }}_{\left( \gamma \right) } {\varvec{X}}_{\left( \gamma \right) } )^{{\prime }}{\varvec{\Sigma }}^{-1}(\mathbf{B}-\mathbf{B}_0 -{\varvec{\Theta }}_{\left( \gamma \right) } {\varvec{X}}_{\left( \gamma \right) } )+{\varvec{\Theta }}_{(\gamma )} {\varvec{H}}_{(\gamma )}^{-1} {\varvec{\Theta }}_{(\gamma )}^{\prime } {\varvec{\Sigma }}^{-1}} \right] } \right\} } \\\propto & {} {\hbox {etr}\left\{ {-\frac{1}{2}\left[ {{\varvec{\Theta }}_{(\gamma )} \left( {{\varvec{X}}_{(\gamma )} {\varvec{X}}_{(\gamma )} ^{{\prime }}+{\varvec{H}}_{(\gamma )}^{-1} } \right) {\varvec{\Theta }}_{(\gamma )}^{\prime } {\varvec{\Sigma }}^{-1}-2{\varvec{\Sigma }}^{-1}(\mathbf{B}-\mathbf{B}_0 ){\varvec{X}}_{(\gamma )} ^{{\prime }}{\varvec{\Theta }}_{(\gamma )}^{\prime } } \right] } \right\} ,} \end{aligned}$$

the full conditional distribution of \({\varvec{\Theta }}_{\left( \gamma \right) } \) is \(\hbox {MN}(\tilde{\varvec{\Theta }} _{\left( \gamma \right) } ,\mathbf{K}_{\left( \gamma \right) }^{-1} ,{\varvec{\Sigma }})\), where \({\varvec{K}}_{(\gamma )} ={\varvec{X}}_{(\gamma )} {\varvec{X}}_{(\gamma )}^{{\prime }} +{\varvec{H}}_{(\gamma )}^{-1} \) and \(\tilde{\varvec{\Theta }} _{(\gamma )} =({\varvec{B}}-\mathbf{B}_0 ){\varvec{X}}_{(\gamma )}^{{\prime }} {\varvec{K}}_{(\gamma )}^{-1} \).

(ix) Since

$$\begin{aligned}&\pi ({\varvec{\Theta }}_{\left( \gamma \right) } , {\varvec{\Sigma }}^{-1},\varvec{\gamma } |\hbox {all others}) \\&\quad \propto |{\varvec{\Sigma }}^{-1}|^{\frac{J}{2}}\hbox {etr}\left\{ {-\frac{1}{2}\left[ {(\mathbf{B}-\mathbf{B}_0 -{\varvec{\Theta }}_{\left( \gamma \right) } {\varvec{X}}_{\left( \gamma \right) } )^{{\prime }}{\varvec{\Sigma }}^{-1}(\mathbf{B}-\mathbf{B}_0 -{\varvec{\Theta }}_{\left( \gamma \right) } {\varvec{X}}_{\left( \gamma \right) } )} \right] } \right\} \\&\qquad \times |{\varvec{\Sigma }}^{-1}|^{\frac{\sum _k \,\gamma _k }{2}}|{\varvec{H}}_{(\gamma )} |^{\frac{-\mathrm{T}}{2}}\hbox { etr}\left\{ {-\frac{1}{2}\left[ {{\varvec{\Theta }}_{(\gamma )} {\varvec{H}}_{(\gamma )}^{-1} {\varvec{\Theta }}_{(\gamma )}^{\prime } {\varvec{\Sigma }}^{-1}} \right] } \right\} \prod \limits _{k=1}^K \,[w^{\gamma _k }(1-w)^{1-\gamma _k }] \\&\qquad \times |{\varvec{\Sigma }}^{-1}|^{\frac{v-\mathrm{T}-1}{2}}\hbox {etr}\left\{ {-\frac{V^{-1}}{2}{\varvec{\Sigma }}^{-1}} \right\} \\&\quad \propto |{\varvec{H}}_{\left( \gamma \right) } |^{\frac{-\mathrm{T}}{2}}|{\varvec{\Sigma }}^{-1}|^{\frac{J+ \sum _k \,\gamma _k +\nu -\mathrm{T}-1}{2}} \\&\qquad \times \hbox {etr}\left\{ {-\frac{1}{2}\left[ {({\varvec{\Theta }}_{\left( \gamma \right) } -\tilde{\varvec{\Theta }} _{(\gamma )} ) {\varvec{K}}_{(\gamma )} ({\varvec{\Theta }}_{(\gamma )}^{\prime } - \tilde{\varvec{\Theta }} _{(\gamma )}^{\prime } ){\varvec{\Sigma }}^{-1}-\tilde{\varvec{\Theta }} _{(\gamma )} {\varvec{K}}_{(\gamma )} \tilde{\varvec{\Theta }} _{(\gamma )}^{\prime } {\varvec{\Sigma }}^{-1}} \right] } \right\} \\&\qquad \times \hbox {etr}\left\{ {-\frac{1}{2}\left[ {\left( {\mathbf{B}-\mathbf{B}_0 } \right) ^{{{\prime }}}{\varvec{\Sigma }}^{-1}\left( {\mathbf{B}-\mathbf{B}_0 } \right) +V^{-1}{\varvec{\Sigma }}^{-1}} \right] } \right\} \prod \limits _{k=1}^K \,[w^{\gamma _k }(1-w)^{1-\gamma _k }], \end{aligned}$$

we first integrate out \({\varvec{\Theta }}_{\left( \gamma \right) } \) to get the distribution \(\pi \left( {{\varvec{\Sigma }}^{-1},\varvec{\gamma } \hbox {|all others except }{\varvec{\Theta }}_{\left( \gamma \right) } } \right) \) which is proportional to

$$\begin{aligned}&(\left| {{\varvec{H}}_{\left( \gamma \right) } } \right| \left| {{\varvec{K}}_{\left( \gamma \right) } } \right| )^{\frac{-\mathrm{T}}{2}}\prod \nolimits _k \,w^{\gamma _k }(1-w)^{1-\gamma _k }|{\varvec{\Sigma }}^{-1}|^{\frac{J+\nu -\mathrm{T}-1}{2}}\hbox {etr}\left\{ -\frac{1}{2}\left[ \left( {\mathbf{B}-\mathbf{B}_0 } \right) \left( {\mathbf{B}-\mathbf{B}_0 } \right) ^{{{\prime }}}\right. \right. \\&\quad \left. \left. +V^{-1}\,\mathbf{I}_\mathrm{T} - \tilde{\varvec{\Theta }} _{(\gamma )} {\varvec{K}}_{(\gamma )} \tilde{\varvec{\Theta }} _{(\gamma )}^{\prime } \right] {\varvec{\Sigma }}^{-1} \right\} . \end{aligned}$$

Then, the required result in (23) is obtained by integrating out \({\varvec{\Sigma }}^{-1}\) from this last expression.

Appendix 2: Proof of Theorem 1

For any generated \({\varvec{A}}\), by applying the QR decomposition method, one can obtain a unique orthogonal matrix \({\varvec{\Gamma }}\) such that \({\varvec{\Gamma }}{\varvec{A}}\) satisfies the identification constraint given in (9). Other identified parameters are then obtained by multiplying \({\varvec{\Gamma }}\) to those parameters (e.g., \({\varvec{\Gamma }}{\varvec{B}})\). Now \(\mathbf{V}_{(\gamma )}\) in (23) can be re-written as

$$\begin{aligned}&\mathbf{V}_{(\gamma )} =V^{-1}\mathbf{I}_\mathrm{T} +({\varvec{B}}-\mathbf{1}^{{{\prime }}} {\otimes }{\varvec{b}}_0 )(\mathbf{I}_\mathrm{J} -{\varvec{X}}_{\left( \gamma \right) }^{{\prime }} {\varvec{K}}_{\left( \gamma \right) }^{-1} {\varvec{X}}_{\left( \gamma \right) } )({\varvec{B}}-\mathbf{1}^{{{\prime }}}{\otimes }{\varvec{b}}_0 )^{{\prime }} \\&\quad =V^{-1}{\varvec{\Gamma }}^{\prime }{\varvec{\Gamma }} +{\varvec{\Gamma }}^{\prime }({\varvec{\Gamma }}{\varvec{B}} -{\varvec{\Gamma }}[\mathbf{1}^{{{\prime }}}{\otimes }{\varvec{b}}_0 ]) (\mathbf{I}_\mathrm{J} -{\varvec{X}}_{\left( \gamma \right) }^{{\prime }} {\varvec{K}}_{\left( \gamma \right) }^{-1} {\varvec{X}}_{\left( \gamma \right) } )({\varvec{\Gamma }}{\varvec{B}} -{\varvec{\Gamma }}[\mathbf{1}^{{{\prime }}}{\otimes }{\varvec{b}}_0 ])^{{\prime }}{\varvec{\Gamma }} \\&\quad ={\varvec{\Gamma }}^{{\prime }}\left[ {V^{-1}\mathbf{I}_\mathrm{T} +\left( {{\varvec{\Gamma }}{\varvec{B}}-\mathbf{1}^{{{\prime }}}{\otimes }{\varvec{\Gamma }}{\varvec{b}}_0 )(\mathbf{I}_\mathrm{J} -{\varvec{X}}_{\left( \gamma \right) }^{{\prime }} {\varvec{K}}_{\left( \gamma \right) }^{-1} {\varvec{X}}_{\left( \gamma \right) } } \right) \left( {{\varvec{\Gamma }}{\varvec{B}} -\mathbf{1}^{{{\prime }}}{\otimes }{\varvec{\Gamma }}{\varvec{b}}_0 } \right) ^{{{\prime }}}} \right] {\varvec{\Gamma }}, \end{aligned}$$

so

$$\begin{aligned}&|\mathbf{V}_{\left( \gamma \right) } |=|{\varvec{\Gamma }}^{{\prime }}\left[ {V^{-1}\mathbf{I}_\mathrm{T} +\left( {{\varvec{\Gamma }}{\varvec{B}}-\mathbf{1}^{{{\prime }}}{\otimes }{\varvec{\Gamma }}{\varvec{b}}_0 )(\mathbf{I}_\mathrm{J} -{\varvec{X}}_{\left( \gamma \right) }^{{\prime }} {\varvec{K}}_{\left( \gamma \right) }^{-1} {\varvec{X}}_{\left( \gamma \right) } } \right) \left( {{\varvec{\Gamma }}{\varvec{B}}-\mathbf{1}^{{{\prime }}}{\otimes }{\varvec{\Gamma }}{\varvec{b}}_0 } \right) ^{{{\prime }}}} \right] {\varvec{\Gamma }}| \\&\quad = {\vert }{\varvec{\Gamma }}^{{\prime }}||V^{-1}\mathbf{I}_\mathrm{T} +\left( {{\varvec{\Gamma }}{\varvec{B}}-\mathbf{1}^{{{\prime }}}{\otimes }{\varvec{\Gamma }}{\varvec{b}}_0 )(\mathbf{I}_\mathrm{J} -{\varvec{X}}_{\left( \gamma \right) }^{{\prime }} {\varvec{K}}_{\left( \gamma \right) }^{-1} {\varvec{X}}_{\left( \gamma \right) } } \right) \left( {{\varvec{\Gamma }}{\varvec{B}}-1^{{{\prime }}}{\otimes }{\varvec{\Gamma }}{\varvec{b}}_0} \right) ^{{{\prime }}}||{\varvec{\Gamma }}| \\&\quad =|V^{-1}\mathbf{I}_\mathrm{T} +\left( {{\varvec{\Gamma }}{\varvec{B}}-\mathbf{1}^{{{\prime }}}{\otimes }{\varvec{\Gamma }}{\varvec{b}}_0 )(\mathbf{I}_\mathrm{J} -{\varvec{X}}_{\left( \gamma \right) }^{{\prime }} {\varvec{K}}_{\left( \gamma \right) }^{-1} {\varvec{X}}_{\left( \gamma \right) } } \right) \left( {{\varvec{\Gamma }}{\varvec{B}}-\mathbf{1}^{{{\prime }}}{\otimes }{\varvec{\Gamma }}{\varvec{b}}_0 } \right) ^{{{\prime }}}|. \end{aligned}$$

Since \(|\mathbf{V}_{\left( \gamma \right) } |\) as well as the remaining terms in (23) are unchanged when the unidentified parameters (e.g., \({\varvec{B}}\)) are replaced by the identified parameters (e.g., \({\varvec{\Gamma }}{\varvec{B}})\), the posterior distribution of \(\varvec{\gamma }\) is unchanged when the substitution is made. Thus, the variable selection results are not affected by the proposed post-processing procedure.

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Fong, D.K.H., DeSarbo, W.S., Chen, Z. et al. A Bayesian Vector Multidimensional Scaling Procedure Incorporating Dimension Reparameterization with Variable Selection. Psychometrika 80, 1043–1065 (2015). https://doi.org/10.1007/s11336-015-9449-x

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