| 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{\mu} and covariance matrix \boldsymbol{\Sigma}.
The conjugate prior is
\pi(\boldsymbol \mu, \boldsymbol \Sigma\mid\boldsymbol m_0,\kappa_0,\nu_0,\boldsymbol \Lambda_0)=
\pi_{\mu}(\boldsymbol \mu|\boldsymbol \Sigma,\boldsymbol m_0,\kappa_0)\pi_{\Sigma}(\boldsymbol \Sigma \mid \nu_0,\boldsymbol \Lambda_0),
\pi_{\mu}(\boldsymbol \mu|\boldsymbol \Sigma,\boldsymbol m_0,\kappa_0) =
\frac{\sqrt{\kappa_0^d}}{\sqrt {(2\pi )^{d}|{\boldsymbol \Sigma }|}} \exp \left(-{\frac {\kappa_0}{2}}(\boldsymbol\mu -{\boldsymbol m_0 })^{\mathrm {T} }{\boldsymbol{\Sigma }}^{-1}(\boldsymbol\mu-{\boldsymbol m_0 })\right),
\qquad \boldsymbol \mu\in\mathcal{R}^d,
\pi_{\Sigma}(\boldsymbol \Sigma\mid \nu_0,\boldsymbol \Lambda_0)= {\frac {\left|{\boldsymbol \Lambda_0 }\right|^{\nu_0 /2}}{2^{\nu_0 d/2}\Gamma _{d}({\frac {\nu_0 }{2}})}}\left|\boldsymbol \Sigma \right|^{-(\nu_0 +d+1)/2}e^{-{\frac {1}{2}}\mathrm {tr} (\boldsymbol \Lambda_0 \boldsymbol \Sigma^{-1})}
, \qquad \boldsymbol \Sigma^2>0,
where mu0 corresponds to \boldsymbol m_0, ka0 corresponds to \kappa_0,
nu0 to \nu_0, and Lam0 to \Lambda_0.
Usage
AM_mix_hyperparams_multinorm(mu0 = NULL, ka0 = NULL, nu0 = NULL, Lam0 = NULL)
Arguments
mu0 |
The hyperparameter |
ka0 |
The hyperparameter |
nu0 |
The hyperparameter |
Lam0 |
The hyperparameter |
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 \nu_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 ()