Summary of learning nu in all considered models
Last updated: 2025-12-02
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In conclusion, the considered models can be ordered as follows (MLE \(\nu\) from large to small. Recall that we want \(\nu\) to be reasonably large in this experiment):
\(\mathcal{IW}(\Psi_{PCA} | \nu \Psi'_{PCA}, \nu + p + 1)\)
\(\mathrm{F}(R_0 | \frac{\nu R'}{N}, N, \nu + 2)\)
\(\mathrm{F}(R_0 | \frac{\nu \Psi_{Bayes}'}{N}, N, \nu + 2)\)
\(\mathrm{F}(R_0 | \frac{\nu \Psi_{PCA}'}{N}, N, \nu + 2)\)
\(\mathcal{IW}(R_0 | \frac{\nu \Psi_{Bayes}'}{N}, \nu + p + 1)\)
\(\mathcal{IW}(R_0 | \nu R', \nu + p + 1)\)
where \(\Psi_{PCA}\) and \(\Psi'_{PCA}\) are the PPCA of the in-sample and out-of-sample population covariance matrix with 99% variance explained, and \[\Psi'_{Bayes} = \dfrac{N' R' + \delta I}{N' + \delta},\] for very small \(\delta\). This corresponds to learning the out-of-sample population using posterior mean in a Bayesian procedure.
I think (2) is simular to (3) since \(\delta\) is small. I like those most because those models are natural to motivate.
In this Markdown, we subsample the data to get in-sample \(R_0\) and out-of-sample \(R'\) and check learned \(\nu\) in all models above.
1. Data preparation and log-likelihood functions
First, load the data:
seed = 10 ## change this to see different experiment
set.seed(seed)
library(susieR)Warning: package 'susieR' was built under R version 4.3.3library(Matrix)
data(N3finemapping)
attach(N3finemapping)
X0 = N3finemapping$X
## getting covariance matrix from the whole sample
## and examine the eigendecomposition to estimate numerical rank
R = cov(X0)
eig <- eigen(R)
plot(eig$values,
main = "Eigenvalues of covariance matrix calculated using all samples",
ylab = "Value",
xlab = "Eigenvalue index")
| Version | Author | Date |
|---|---|---|
| 7edf83c | dodat97 | 2025-12-02 |
n0 = dim(X0)[1]
p0 = dim(X0)[2]
snp_total = p0We proceed to split the data into half and look at the heatmap of the covariance matrices of two sub-samples.
#### randomly split the data into half
#### randomly select p consecutive SNPs where p < n so IW is proper
p = 100
# Start from a random point on the genome
indx_start = sample(1: (snp_total - p), 1)
X = X0[, indx_start:(indx_start + p -1)]
# View(cor(X)[1:10, 1:10])
## sub-sample into two
out_sample_size = n0 / 2
out_sample = sample(1:n0, out_sample_size)
X_out = X[out_sample, ]
X_in = X[setdiff(1:n0, out_sample), ]
rm_p = c(which(diag(cov(X_in))==0), which(diag(cov(X_out))==0))
indx_p = setdiff(1:p, rm_p)
X_in = X_in[, indx_p]
X_out = X_out[, indx_p]
## out-sample LD matrix
p = length(indx_p)
Rp = cov(X_out)
R0 = cov(X_in)
library(ggplot2)
library(reshape2)
df1 <- melt(R0)
df2 <- melt(Rp)
N_in = nrow(X_in)
N_out = nrow(X_out)
p1 <- ggplot(df1, aes(Var1, Var2, fill = value)) +
geom_tile() +
scale_fill_gradient2(low="blue", mid="white", high="red") +
coord_fixed() +
ggtitle(sprintf("In-sample Cov, %d samples", nrow(X_in)))
p2 <- ggplot(df2, aes(Var1, Var2, fill = value)) +
geom_tile() +
scale_fill_gradient2(low="blue", mid="white", high="red") +
coord_fixed() +
ggtitle(sprintf("Out-of-sample Cov, %d samples", nrow(X_out)))
library(gridExtra)
grid.arrange(p1, p2, ncol = 2)
Log-likelihood functions:
## Matrix F-distribution likelihood
#### log F(R0 | nu * Rp / N, N, nu + 2)
log_multigamma_vec <- function(a, p) {
# vectorized multivariate gamma
shifts <- (1 - seq_len(p)) / 2
# sum over j, but broadcasting a over j
(p*(p-1)/4)*log(pi) +
rowSums(lgamma(outer(a, shifts, "+")))
}
log_F <- function(R0, Rp, N, nu_vec) {
p <- nrow(R0)
jitter = 1e-8
R0 = R0 + jitter * diag(p)
Rp = Rp + jitter * diag(p)
# Precompute expensive shared quantities
logdet_nu_Rp_over_N <- (determinant(Rp, logarithm = TRUE)$modulus
+ p * log(nu_vec)
- p * log(N))
logdetR0 <- determinant(R0, logarithm = TRUE)$modulus
# lambda_vec <- eigen(solve(Rp, R0))$values
# lambda_over_nu = tcrossprod(lambda_vec, N / nu_vec)
# logdet_I_plus_RR <- colSums(log(1 + lambda_over_nu))
# llhs = (log_multigamma_vec((N + nu_vec + p + 1) / 2, p)
# - log_multigamma_vec(N / 2, p)
# - log_multigamma_vec((nu_vec + p + 1) / 2, p)
# - .5 * N * logdet_nu_Rp_over_N
# + .5 * (N - p - 1) * logdetR0
# - .5 * (N + nu_vec + p + 1) * logdet_I_plus_RR)
logdet_Rplus_Rp = rep(0, length(nu_vec))
for (idx in 1:length(nu_vec)){
nu = nu_vec[idx]
logdet_Rplus_Rp[idx] <- determinant(R0 + nu * Rp / N, logarithm = TRUE)$modulus
}
llhs = (log_multigamma_vec((N + nu_vec + p + 1) / 2, p)
- log_multigamma_vec(N / 2, p)
- log_multigamma_vec((nu_vec + p + 1) / 2, p)
+ .5 * (nu_vec + p + 1) * logdet_nu_Rp_over_N
+ .5 * (N - p - 1) * logdetR0
- .5 * (N + nu_vec + p + 1) * logdet_Rplus_Rp)
as.numeric(llhs)
}
log_iw <- function(R0, Rp, nu_vec) {
p <- nrow(R0)
jitter = 1e-12
R0 = R0 + jitter * diag(p)
Rp = Rp + jitter * diag(p)
# Precompute expensive shared quantities
logdet_nu_Rp <- determinant(Rp, logarithm = TRUE)$modulus + p * log(nu_vec)
logdetR0 <- determinant(R0, logarithm = TRUE)$modulus
tr_term <- nu_vec * sum(t(Rp) * solve(R0))
llhs = (.5 * (nu_vec + p + 1) * logdet_nu_Rp
- .5 * (nu_vec + p + 1) * p * log(2)
- log_multigamma_vec((nu_vec + p + 1) / 2, p)
- .5 * (nu_vec + 2 * (p + 1)) * logdetR0
- .5 * tr_term)
as.numeric(llhs)
}2. Comparing \(\nu\) learned in the considered models
Model 1
\[\mathcal{IW}(\Psi_{PCA} | \nu \Psi'_{PCA}, \nu + p + 1)\] This is not really a model for \(R_0\), but it gives largest \(\nu\).
eig <- eigen(Rp)
eig_cumsum = cumsum(eig$values)
percent_explained = .99
eig_cumsum = cumsum(eig$values)
q = sum(eig_cumsum < percent_explained * eig_cumsum[p])
sprintf("%d first principle components explain %.1f percent of variance", q, percent_explained*100)[1] "40 first principle components explain 99.0 percent of variance"lambda <- eig$values
U <- eig$vectors
sigma2_est <- mean(lambda[(q+1):p])
L_diag <- sqrt(lambda[1:q] - sigma2_est)
# L_diag <- sqrt(lambda[1:q])
W_ppca <- U[,1:q] %*% diag(L_diag)
Rp_PCA <- W_ppca %*% t(W_ppca) + sigma2_est * diag(p)
eig <- eigen(R0)
lambda <- eig$values
U <- eig$vectors
sigma2_est <- mean(lambda[(q+1):p])
L_diag <- sqrt(lambda[1:q] - sigma2_est)
# L_diag <- sqrt(lambda[1:q])
W_ppca <- U[,1:q] %*% diag(L_diag)
R0_PCA <- W_ppca %*% t(W_ppca) + sigma2_est * diag(p)
nu_vec = c(1:100)
llhs = log_iw(R0_PCA, Rp_PCA, nu_vec)
# llhs = log_iw(R0_PCA, Rp, nu_vec)
# llhs = log_iw(R0, Rp_PCA, nu_vec)
plot(nu_vec, llhs, xlab = "nu value", ylab = "log-likelihood")
print(nu_vec[which.max(llhs)])[1] 98Model 2
\[\mathrm{F}(R_0 | \frac{\nu R'}{N}, N, \nu + 2)\] This model corresponds to the hierarchy \[R_0 | \Psi \sim \mathcal{W}(\Psi / N, N), \quad \Psi | R' \sim \mathcal{IW}(\nu R', \nu + p + 1).\]
N = nrow(X_in)
nu_vec = c(1:100)
llhs = log_F(R0, Rp, N, nu_vec)
plot(nu_vec, llhs, xlab = "nu value", ylab = "log-likelihood of matrix F-distribution")
print(paste0("Optimal nu is ", nu_vec[which.max(llhs)]))[1] "Optimal nu is 32"Model 3
\[\mathrm{F}(R_0 | \frac{\nu \Psi'}{N}, N, \nu + 2)\] This model corresponds to the hierarchy \[R_0 | \Psi \sim \mathcal{W}(\Psi / N, N), \quad \Psi | \Psi' \sim \mathcal{IW}(\nu \Psi', \nu + p + 1),\] and \(\Psi' = (N' R' + \delta I) / (N' + \delta)\) being the posterior mean of the estimated population covariance matrix.
N = nrow(X_in)
Np = nrow(X_out)
nu_vec = c(1:100)
delta = 1e-6
Psi_p = (Np * Rp + delta * diag(p)) / (Np + delta)
llhs = log_F(R0, Psi_p, N, nu_vec)
plot(nu_vec, llhs, xlab = "nu value", ylab = "log-likelihood of matrix F-distribution")
print(paste0("Optimal nu is ", nu_vec[which.max(llhs)]))[1] "Optimal nu is 32"Model 4
\[\mathrm{F}(R_0 | \frac{\nu \Psi_{PCA}'}{N}, N, \nu + 2)\]
nu_vec = c(1:100)
llhs = log_F(R0, Rp_PCA, N, nu_vec)
plot(nu_vec, llhs, xlab = "nu value", ylab = "log-likelihood of matrix F-distribution")
print(paste0("Optimal nu is ", nu_vec[which.max(llhs)]))[1] "Optimal nu is 18"Model 5
\[\mathrm{IW}(R_0 | \nu \Psi_{Bayes}', \nu + p + 1)\]
nu_vec = c(1:100)
llhs = log_iw(R0, Psi_p, nu_vec)
plot(nu_vec, llhs, xlab = "nu value", ylab = "log-likelihood")
print(nu_vec[which.max(llhs)])[1] 1Model 6
Model 6: This is our original easiest model, but the estimated \(\nu\) in this model is very small \[\mathrm{IW}(R_0 | \nu R', \nu + p + 1)\]
nu_vec = c(1:100)
llhs = log_iw(R0, Rp, nu_vec)
plot(nu_vec, llhs, xlab = "nu value", ylab = "log-likelihood")
print(nu_vec[which.max(llhs)])[1] 1
sessionInfo()R version 4.3.2 (2023-10-31)
Platform: aarch64-apple-darwin20 (64-bit)
Running under: macOS 15.6.1
Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/lib/libRlapack.dylib; LAPACK version 3.11.0
locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
time zone: America/Chicago
tzcode source: internal
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
[1] gridExtra_2.3 reshape2_1.4.5 ggplot2_4.0.1 Matrix_1.6-1.1
[5] susieR_0.14.2 workflowr_1.7.2
loaded via a namespace (and not attached):
[1] sass_0.4.10 generics_0.1.4 stringi_1.8.7 lattice_0.22-7
[5] digest_0.6.39 magrittr_2.0.4 evaluate_1.0.5 grid_4.3.2
[9] RColorBrewer_1.1-3 fastmap_1.2.0 plyr_1.8.9 rprojroot_2.1.1
[13] jsonlite_2.0.0 processx_3.8.6 whisker_0.4.1 reshape_0.8.10
[17] mixsqp_0.3-54 ps_1.9.1 promises_1.5.0 httr_1.4.7
[21] scales_1.4.0 jquerylib_0.1.4 cli_3.6.5 rlang_1.1.6
[25] crayon_1.5.3 withr_3.0.2 cachem_1.1.0 yaml_2.3.11
[29] otel_0.2.0 tools_4.3.2 dplyr_1.1.4 httpuv_1.6.16
[33] vctrs_0.6.5 R6_2.6.1 matrixStats_1.5.0 lifecycle_1.0.4
[37] git2r_0.36.2 stringr_1.6.0 fs_1.6.6 irlba_2.3.5.1
[41] pkgconfig_2.0.3 callr_3.7.6 pillar_1.11.1 bslib_0.9.0
[45] later_1.4.4 gtable_0.3.6 glue_1.8.0 Rcpp_1.1.0
[49] xfun_0.54 tibble_3.3.0 tidyselect_1.2.1 rstudioapi_0.17.1
[53] knitr_1.50 farver_2.1.2 htmltools_0.5.8.1 labeling_0.4.3
[57] rmarkdown_2.30 compiler_4.3.2 getPass_0.2-4 S7_0.2.1