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Rapport Année : 2021

Predictive distributions of random variables following a multivariate Gaussian distribution with Normal Inverse Wishart prior

Résumé

The Gaussian distribution is a popular choice of density for the modeling of continuous observations, often combined with a specific prior in the context of Bayesian inference. When this prior is conjugate (e.g., the normal-gamma inverse distribution in the univariate case), it can be proved that the posterior and predictive distributions have analytically tractable expressions. While these derivations are readily available in the literature for the univariate case, the multivariate case derivations are significantly harder to find. This short technical note proposes a detailed derivation of these closed-forms in the multivariate case with a conjugate Normal Inverse Wishart prior. Keywords Bayesian Inference • Predictive Distributions • Multivariate Gaussian • Normal Inverse Wishart The Gaussian distribution is a popular choice of density for the modeling of continuous observations, often combined with a specific prior in the context of Bayesian inference. When this prior is conjugate (e.g., the normal-gamma inverse distribution in the univariate case), it can be proved that the posterior and predictive distributions have analytically tractable expressions. The most cited source [1] (e.g., cited in [2, 3]) proves the equivalence in the univariate case. However, the multivariate case derivations are significantly harder to find. In the following we propose a detailed derivation of the multivariate distribution, based on the exponential family expressions of the Gaussian and Normal Inverse Wishart distributions.
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Dates et versions

hal-03962723 , version 1 (30-01-2023)

Licence

Paternité - Pas d'utilisation commerciale - Pas de modification

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  • HAL Id : hal-03962723 , version 1

Citer

Etienne Goffinet, Mustapha Lebbah, Hanane Azzag, Loïc Giraldi. Predictive distributions of random variables following a multivariate Gaussian distribution with Normal Inverse Wishart prior: Technical report. LIPN. 2021. ⟨hal-03962723⟩
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