This function is largely a more user friendly wrapper around
optimMaltipooCollapsed and
uncollapsePibble.
See details for model specification.
Notation: N is number of samples,
D is number of multinomial categories, Q is number
of covariates, P is the number of variance components
iter is the number of samples of eta (e.g.,
the parameter n_samples in the function
optimPibbleCollapsed)
Usage
maltipoo(
Y = NULL,
X = NULL,
upsilon = NULL,
Theta = NULL,
U = NULL,
Xi = NULL,
init = NULL,
ellinit = NULL,
pars = c("Eta", "Lambda", "Sigma"),
...
)Arguments
- Y
D x N matrix of counts (if NULL uses priors only)
- X
Q x N matrix of covariates (design matrix) (if NULL uses priors only, must be present to sample Eta)
- upsilon
dof for inverse wishart prior (numeric must be > D) (default: D+3)
- Theta
(D-1) x Q matrix of prior mean for regression parameters (default: matrix(0, D-1, Q))
- U
a PQ x Q matrix of stacked variance components (each of dimension Q x Q)
- Xi
(D-1)x(D-1) prior covariance matrix (default: ALR transform of diag(1)*(upsilon-D)/2 - this is essentially iid on "base scale" using Aitchison terminology)
- init
(D-1) x Q initialization for Eta for optimization
- ellinit
P vector initialization values for ell for optimization
- pars
character vector of posterior parameters to return
- ...
arguments passed to
optimPibbleCollapsedanduncollapsePibble
Details
the full model is given by: $$Y_j \sim Multinomial(Pi_j)$$ $$Pi_j = Phi^{-1}(Eta_j)$$ $$Eta \sim MN_{D-1 x N}(Lambda*X, Sigma, I_N)$$ $$Lambda \sim MN_{D-1 x Q}(Theta, Sigma, Gamma)$$ $$Gamma = e^{ell_1} U_1 + ... + e^{ell_P} U_P$$ $$Sigma \sim InvWish(upsilon, Xi)$$
Where \(A = (I_N + X * Gamma * X')^{-1}\), \(K^{-1} = Xi\) is a (D-1)x(D-1) covariance matrix, \(U_1\) is a Q x Q covariance matrix (a variance component), \(e^{ell_i}\) is a scale for that variance component and \(Phi^{-1}\) is ALRInv_D transform.
Default behavior is to use MAP estimate for uncollaping collapsed maltipoo model if laplace approximation is not preformed.
Parameters ell are treated as fixed and estimated by MAP estimation.