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A Bayesian method for multinomial probit model

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Abstract

The independence of irrelevant alternatives (IIA) property states that the ratio of any two choice probabilities in a set of alternatives is independent of the presence or absence of other alternatives. In the modeling of multinomial data, the IIA is not feasible. In this situation, the multinomial probit (MNP) model is a type of discrete choice model that is commonly used. Due to the identifiability problem and the positive-definiteness constraint, modeling the covariance matrix in the MNP is difficult. All existing methods use unidentifiable parameters in the covariance matrix to solve the unidentifiability problem and improve the rate of convergence of a data augmentation algorithm. These methods also use the inverse Wishart distribution, which is frequently insufficient (Barnard et al. Stat Sin 10(4):1281–1311, 2000). We employed variance-correlation decomposition to decompose the identifiable covariance matrix into standard deviations and a correlation matrix instead of using the unidentifiable covariance matrix. Hypersphere decomposition was also used to decompose the correlation matrix. Thus, the estimated covariance matrix satisfied the positive definiteness constraint. The performance of our proposed model was illustrated using a detergent dataset from market research.

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Acknowledgements

This project was supported by Basic Science Research Program through the National Research Foundation of Korea (KRF) funded by the Korean government (NRF-2022R1A2C1002752) for Keunbaik Lee.

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Authors and Affiliations

  1. Department of Statistics, Sungkyunkwan University, Seoul, Korea

    Donghyun Koo, Chanmin Kim & Keunbaik Lee

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  1. Donghyun Koo
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  2. Chanmin Kim
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  3. Keunbaik Lee
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Correspondence to Keunbaik Lee.

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Koo, D., Kim, C. & Lee, K. A Bayesian method for multinomial probit model. J. Korean Stat. Soc. 52, 265–281 (2023). https://doi.org/10.1007/s42952-022-00200-5

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