ABSTRACT
The first, and perhaps most natural, departure from the Gaussian mixture model is the mixture of multivariate t-distributions. McLachlan and Peel (1998) motivate the multivariate t-distribution as a heavy-tailed alternative to the multivariate Gaussian distribution by first considering the Gaussian scale mixture model ( 1 − ∈ ) ϕ ( x | μ , Σ ) + ∈ ϕ ( x | μ , c Σ ) , ( 5.1 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315373577/3ff373dc-6a7d-4d8f-9bc3-d8ed9230a904/content/eqn5_1.jpg"/> where c is large, ∈ is small, and φ(x | μ, Σ) is the density of a multivariate Gaussian distribution with mean μ and covariance matrix Σ. Now (5.1) can be written ∫ ϕ ( x | μ , Σ / w ) d H ( ω ) , ( 5.2 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315373577/3ff373dc-6a7d-4d8f-9bc3-d8ed9230a904/content/eqn5_2.jpg"/> where H is the probability distribution placing mass (1 - ∈) at the point w = 1 and mass ∈ at the point w = 1/c. Then the multivariate t-distribution is obtained when H in (5.2) is replaced by the probability density of a random variable W ~ gamma(v/2,v/2), where v denotes degrees of freedom (McLachlan and Peel, 1998). Note that the gamma(a, b) density is h ( w | α , β ) = β α w α − 1 exp { − β w } Γ ( α ) , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315373577/3ff373dc-6a7d-4d8f-9bc3-d8ed9230a904/content/ueqn5_1.jpg"/> for w > 0 with parameters α , β ∈ ℝ + https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315373577/3ff373dc-6a7d-4d8f-9bc3-d8ed9230a904/content/ineqn5_1.jpg"/> . The density for the multivariate t-distribution is f t ( x | μ , Σ , v ) = Γ ( [ v + p ] / 2 ) | Σ | − 1 / 2 ( π v ) p / 2 Γ ( v / 2 ) [ 1 + δ ( x , μ | Σ ) / v ] ( v + p ) / 2 , ( 5.3 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315373577/3ff373dc-6a7d-4d8f-9bc3-d8ed9230a904/content/eqn5_3.jpg"/> with mean μ , scale matrix Σ, and degrees of freedom v, and where δ ( x , μ | Σ ) = ( x − μ ) ′ Σ − 1 ( x − μ ) ( 5.4 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315373577/3ff373dc-6a7d-4d8f-9bc3-d8ed9230a904/content/eqn5_4.jpg"/> 80is the squared Mahalanobis distance between x and μ ,. The density for a mixture of multivariate t-distributions is given by f ( x | ϑ ) = ∑ g = 1 G π g f t ( x | μ g , Σ g , v g ) . ( 5.5 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315373577/3ff373dc-6a7d-4d8f-9bc3-d8ed9230a904/content/eqn5_5.jpg"/>
