uncollapsePibble.RdSee details for model. Should likely be called following
optimPibbleCollapsed. Notation: N is number of samples,
D is number of multinomial categories, Q is number
of covariates, iter is the number of samples of eta (e.g.,
the parameter n_samples in the function optimPibbleCollapsed)
uncollapsePibble(eta, X, Theta, Gamma, Xi, upsilon, seed, ret_mean = FALSE, ncores = -1L)
| eta | array of dimension (D-1) x N x iter (e.g., |
|---|---|
| X | matrix of covariates of dimension Q x N |
| Theta | matrix of prior mean of dimension (D-1) x Q |
| Gamma | covariance matrix of dimension Q x Q |
| Xi | covariance matrix of dimension (D-1) x (D-1) |
| upsilon | scalar (must be > D) degrees of freedom for InvWishart prior |
| seed | seed to use for random number generation |
| ret_mean | if true then uses posterior mean of Lambda and Sigma corresponding to each sample of eta rather than sampling from posterior of Lambda and Sigma (useful if Laplace approximation is not used (or fails) in optimPibbleCollapsed) |
| ncores | (default:-1) number of cores to use, if ncores==-1 then uses default from OpenMP typically to use all available cores. |
List with components
Lambda Array of dimension (D-1) x Q x iter (posterior samples)
Sigma Array of dimension (D-1) x (D-1) x iter (posterior samples)
Timer
Notation: Let Z_j denote the J-th row of a matrix Z. While the collapsed model is given by: $$Y_j ~ Multinomial(Pi_j)$$ $$Pi_j = Phi^{-1}(Eta_j)$$ $$Eta ~ T_{D-1, N}(upsilon, Theta*X, K, A)$$ Where A = I_N + X * Gamma * X', K = Xi is a (D-1)x(D-1) covariance matrix, Gamma is a Q x Q covariance matrix, and Phi^-1 is ALRInv_D transform.
The uncollapsed model (Full pibble model) is given by:
$$Y_j ~ Multinomial(Pi_j)$$
$$Pi_j = Phi^{-1}(Eta_j)$$
$$Eta ~ MN_{D-1 x N}(Lambda*X, Sigma, I_N)$$
$$Lambda ~ MN_{D-1 x Q}(Theta, Sigma, Gamma)$$
$$Sigma ~ InvWish(upsilon, Xi)$$
This function provides a means of sampling from the posterior distribution of
Lambda and Sigma given posterior samples of Eta from
the collapsed model.
JD Silverman K Roche, ZC Holmes, LA David, S Mukherjee. Bayesian Multinomial Logistic Normal Models through Marginally Latent Matrix-T Processes. 2019, arXiv e-prints, arXiv:1903.11695
sim <- pibble_sim() # Fit model for eta fit <- optimPibbleCollapsed(sim$Y, sim$upsilon, sim$Theta%*%sim$X, sim$KInv, sim$AInv, random_pibble_init(sim$Y)) # Finally obtain samples from Lambda and Sigma fit2 <- uncollapsePibble(fit$Samples, sim$X, sim$Theta, sim$Gamma, sim$Xi, sim$upsilon, seed=2849)