attempt fitting matrix-normal distribution using Kronecker product and inverse Wishart
Timothée Flutre
I am encountering an error when fitting a matrix-normal distribution with this simple example (see the attached R file):
- Y ~ MN(M, U, V)
- Y is NxT; M is NxT; U is NxN; V is TxT where N is the number of samples and T the number of traits
- here, I assume M to be fixed and full of 0, and U to be fixed and equal to the identity
- goal: estimate V (for the sake of simplicity, I'm using an inverse Wishart as prior, even though the STAN manual gives better alternatives)
- trick: vec(Y) ~ MVN(vec(M), V kx U) where "kx" is the Kronecker product
Here is the model in STAN syntax (thanks to code from Tanner Sorensen via a message sent on 30/01/2015):
data {
int<lower=0> N;
int<lower=0> T;
vector[N*T] vecY;
vector[N*T] vecM;
matrix[N,N] U;
matrix[T,T] S;
}
parameters {
matrix[T,T] V;
}
model {
matrix[T*N,T*N] Sigma;
V ~ inv_wishart(T+1, S);
for(ti in 1:T)
for(tj in 1:T)
for(ni in 1:N)
for(nj in 1:N)
Sigma[N*(ti-1)+ni, N*(tj-1)+nj] <- V[ti,tj] * U[ni,nj];
vecY ~ multi_normal(vecM, Sigma);
}
But when I use it, I get:
[1] "Rejecting initial value:"
[2] " Error evaluating the log probability at the initial value."
[3] "Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:"
[4] "Exception thrown at line 14: stan::math::inv_wishart_log: LDLT_Factor of random variable is not positive definite. last conditional variance is 3.66872."
[5] "If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,"
[6] "but if this warning occurs often then your model may be either severely ill-conditioned or misspecified."
[7] "Rejecting initial value:"
[8] " Error evaluating the log probability at the initial value."
[9] "Initialization between (-2, 2) failed after 100 attempts. "
[10] " Try specifying initial values, reducing ranges of constrained values, or reparameterizing the model."
The problem seems to come from the inverse Wishart prior, but why?
ps: when I draw a random sample from MCMCpack::riwish(v=T+1, S=diag(T)) everything seems fine.
R version 3.2.2 (2015-08-14)
Platform: x86_64-pc-linux-gnu (64-bit)
Running under: Ubuntu 14.04.3 LTS
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
[1] MCMCpack_1.3-3 coda_0.18-1 rstan_2.8.0 ggplot2_1.0.1 Rcpp_0.12.2
[6] MASS_7.3-44
loaded via a namespace (and not attached):
[1] lattice_0.20-33 codetools_0.2-14 digest_0.6.8 grid_3.2.2
[5] plyr_1.8.3 gtable_0.1.2 stats4_3.2.2 magrittr_1.5
[9] scales_0.3.0 stringi_1.0-1 reshape2_1.4.1 proto_0.3-10
[13] tools_3.2.2 stringr_1.0.0 munsell_0.4.2 compiler_3.2.2
[17] inline_0.3.14 colorspace_1.2-6 gridExtra_2.0.0
Daniel Lee
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<test_matrix-normal.txt>