See details for model. Should likely be followed by function
uncollapsePibble. Notation: N is number of samples,
D is number of multinomial categories, and Q is number
of covariates.
optimMaltipooCollapsed(
Y,
upsilon,
Theta,
X,
KInv,
U,
init,
ellinit,
n_samples = 2000L,
calcGradHess = TRUE,
b1 = 0.9,
b2 = 0.99,
step_size = 0.003,
epsilon = 1e-06,
eps_f = 1e-10,
eps_g = 1e-04,
max_iter = 10000L,
verbose = FALSE,
verbose_rate = 10L,
decomp_method = "cholesky",
eigvalthresh = 0,
jitter = 0
)D x N matrix of counts
(must be > D)
D-1 x Q matrix the prior mean for regression coefficients
Q x N matrix of covariates
D-1 x D-1 symmetric positive-definite matrix
a PQxQ matrix of stacked variance components
D-1 x N matrix of initial guess for eta used for optimization
P vector of initial guess for ell used for optimization
number of samples for Laplace Approximation (=0 very fast as no inversion or decomposition of Hessian is required)
if n_samples=0 should Gradient and Hessian still be calculated using closed form solutions?
(ADAM) 1st moment decay parameter (recommend 0.9) "aka momentum"
(ADAM) 2nd moment decay parameter (recommend 0.99 or 0.999)
(ADAM) step size for descent (recommend 0.001-0.003)
(ADAM) parameter to avoid divide by zero
(ADAM) normalized function improvement stopping criteria
(ADAM) normalized gradient magnitude stopping criteria
(ADAM) maximum number of iterations before stopping
(ADAM) if true will print stats for stopping criteria and iteration number
(ADAM) rate to print verbose stats to screen
decomposition of hessian for Laplace approximation 'eigen' (more stable-slightly, slower) or 'cholesky' (less stable, faster, default)
threshold for negative eigenvalues in decomposition of negative inverse hessian (should be <=0)
(default: 0) if >0 then adds that factor to diagonal of Hessian before decomposition (to improve matrix conditioning)
List containing (all with respect to found optima)
LogLik - Log Likelihood of collapsed model (up to proportionality constant)
Gradient - (if calcGradHess=true)
Hessian - (if calcGradHess=true) of the POSITIVE log posterior
Pars - Parameter value of eta
Samples - (D-1) x N x n_samples array containing posterior samples of eta based on Laplace approximation (if n_samples>0)
VCScale - value of e^ell_i at optima
logInvNegHessDet - the log determinant of the covariacne of the Laplace approximation, useful for calculating marginal likelihood
Notation: Let Z_j denote the J-th row of a matrix Z. Model: $$Y_j \sim Multinomial(Pi_j)$$ $$Pi_j = Phi^{-1}(Eta_j)$$ $$Eta \sim T_{D-1, N}(upsilon, Theta\\*X, K, A)$$
Where A = (I_N + e^ell_1\*X\*U_1\*X' + ... + e^ell_P\*X\*U_P\*X' ), K is a D-1xD-1 covariance and Phi is ALRInv_D transform.
Gradient and Hessian calculations are fast as they are computed using closed form solutions. That said, the Hessian matrix can be quite large [N\(D-1) x N\(D-1)] and storage may be an issue.
Note: Warnings about large negative eigenvalues can either signal
that the optimizer did not reach an optima or (more commonly in my experience)
that the prior / degrees of freedom for the covariance (given by parameters
upsilon and KInv) were too specific and at odds with the observed data.
If you get this warning try the following.
Try restarting the optimization using a different initial guess for eta
Try decreasing (or even increasing)step_size (by increments of 0.001 or 0.002)
and increasing max_iter parameters in optimizer. Also can try
increasing b1 to 0.99 and decreasing eps_f by a few orders
of magnitude
Try relaxing prior assumptions regarding covariance matrix. (e.g., may want
to consider decreasing parameter upsilon closer to a minimum value of
D)
Try adding small amount of jitter (e.g., set jitter=1e-5) to address
potential floating point errors.
S. Ruder (2016) An overview of gradient descent optimization algorithms. arXiv 1609.04747