AM_mix_hyperparams_multinorm {AntMAN}R Documentation

multivariate Normal mixture hyperparameters

Description

Generate a configuration object that specifies a multivariate Normal mixture kernel, where users can specify the hyperparameters for the conjugate prior of the multivariate Normal mixture. We assume that the data are d-dimensional vectors \boldsymbol{y}_i, where \boldsymbol{y}_i are i.i.d Normal random variables with mean \boldsymbol{μ} and covariance matrix \boldsymbol{Σ}. The conjugate prior is

π(\boldsymbol μ, \boldsymbol Σ\mid\boldsymbol m_0,κ_0,ν_0,\boldsymbol Λ_0)= π_{μ}(\boldsymbol μ|\boldsymbol Σ,\boldsymbol m_0,κ_0)π_{Σ}(\boldsymbol Σ \mid ν_0,\boldsymbol Λ_0),

π_{μ}(\boldsymbol μ|\boldsymbol Σ,\boldsymbol m_0,κ_0) = \frac{√{κ_0^d}}{√ {(2π )^{d}|{\boldsymbol Σ }|}} \exp ≤ft(-{\frac {κ_0}{2}}(\boldsymbolμ -{\boldsymbol m_0 })^{\mathrm {T} }{\boldsymbol{Σ }}^{-1}(\boldsymbolμ-{\boldsymbol m_0 })\right), \qquad \boldsymbol μ\in\mathcal{R}^d,

π_{Σ}(\boldsymbol Σ\mid ν_0,\boldsymbol Λ_0)= {\frac {≤ft|{\boldsymbol Λ_0 }\right|^{ν_0 /2}}{2^{ν_0 d/2}Γ _{d}({\frac {ν_0 }{2}})}}≤ft|\boldsymbol Σ \right|^{-(ν_0 +d+1)/2}e^{-{\frac {1}{2}}\mathrm {tr} (\boldsymbol Λ_0 \boldsymbol Σ^{-1})} , \qquad \boldsymbol Σ^2>0,

where mu0 corresponds to \boldsymbol m_0, ka0 corresponds to κ_0, nu0 to ν_0, and Lam0 to Λ_0.

Usage

AM_mix_hyperparams_multinorm(mu0 = NULL, ka0 = NULL, nu0 = NULL, Lam0 = NULL)

Arguments

mu0

The hyperparameter \boldsymbol m_0.

ka0

The hyperparameter κ_0.

nu0

The hyperparameter ν_0.

Lam0

The hyperparameter Λ_0.

Details

Default is (mu0=c(0,..,0), ka0=1, nu0=Dim+2, Lam0=diag(Dim)) with Dim is the dimension of the data y. We advise the user to set ν_0 equal to at least the dimension of the data, Dim, plus 2.

Value

An AM_mix_hyperparams object. This is a configuration list to be used as mix_kernel_hyperparams argument for AM_mcmc_fit.

Examples

AM_mix_hyperparams_multinorm ()

[Package AntMAN version 1.1.0 Index]