Efficient profile likelihood and marginal posteriors when nuisance parameters are those of linear regression models.
Maintainer: Martin Lysy mlysy@uwaterloo.ca
Authors:
Bryan Yates
Consider a model \(p(\boldsymbol{Y} \mid \boldsymbol{B}, \boldsymbol{\Sigma}, \boldsymbol{\theta})\) of the form $$ \boldsymbol{Y} \sim \textrm{Matrix-Normal}(\boldsymbol{X}(\boldsymbol{\theta})\boldsymbol{B}, \boldsymbol{V}(\boldsymbol{\theta}), \boldsymbol{\Sigma}), $$ where \(\boldsymbol{Y}_{n \times q}\) is the response matrix, \(\boldsymbol{X}(\theta)_{n \times p}\) is a covariate matrix which depends on \(\boldsymbol{\theta}\), \(\boldsymbol{B}_{p \times q}\) is the coefficient matrix, \(\boldsymbol{V}(\boldsymbol{\theta})_{n \times n}\) and \(\boldsymbol{\Sigma}_{q \times q}\) are the between-row and between-column variance matrices, and (suppressing the dependence on \(\boldsymbol{\theta}\)) the Matrix-Normal distribution is defined by the multivariate normal distribution \( \textrm{vec}(\boldsymbol{Y}) \sim \mathcal{N}(\textrm{vec}(\boldsymbol{X}\boldsymbol{B}), \boldsymbol{\Sigma} \otimes \boldsymbol{V}), \) where \(\textrm{vec}(\boldsymbol{Y})\) is a vector of length \(nq\) stacking the columns of of \(\boldsymbol{Y}\), and \(\boldsymbol{\Sigma} \otimes \boldsymbol{V}\) is the Kronecker product.
The model above is referred to as a Linear Model with Nuisance parameters (LMN) \((\boldsymbol{B}, \boldsymbol{\Sigma})\), with parameters of interest \(\boldsymbol{\theta}\). That is, the LMN package provides tools to efficiently conduct inference on \(\boldsymbol{\theta}\) first, and subsequently on \((\boldsymbol{B}, \boldsymbol{\Sigma})\), by Frequentist profile likelihood or Bayesian marginal inference with a Matrix-Normal Inverse-Wishart (MNIW) conjugate prior on \((\boldsymbol{B}, \boldsymbol{\Sigma})\).
Useful links:
Report bugs at https://github.com/mlysy/LMN/issues