Jointly optimize nu and delta
Last updated: 2026-01-04
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Knit directory: Improved_LD_SuSiE/
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Consider only model 3 and 5 from Markdown “Summary of learning nu in all considered models” and now we vary both \(\nu\) and \(\delta\).
(Model 3) \(\mathrm{F}(R_0 | \frac{\nu \Psi'}{N}, N, \nu + 2)\)
(Model 5) \(\mathcal{IW}(R_0 | \frac{\nu \Psi'}{N}, \nu + p + 1)\)
where \(\Psi' = \dfrac{\nu R' + \delta \text{diag}(R_0)}{\nu + \delta}\).
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")
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 F model with various delta
Sorry In this simulation \(\Psi' = \dfrac{N' R' + \text{diag}(R_0)}{N' + \delta}\)
This is not exactly setting \(\Psi' = \dfrac{\nu R' + \text{diag}(R_0)}{\nu' + \delta}\) as we want. But I found it still gives some more intuition.
N = nrow(X_in)
Np = nrow(X_out)
nu_vec = c(1:100)
delta = 1e-6
Psi_p = (Np * Rp + delta * diag(R0)) / (Np + delta)
llhs = log_F(R0, Psi_p, N, nu_vec)
plot(nu_vec, llhs, xlab = paste0("nu value; setting delta = ", delta), ylab = "log-likelihood of matrix F-distribution")
print(paste0("Optimal nu is ", nu_vec[which.max(llhs)]))[1] "Optimal nu is 32"N = nrow(X_in)
Np = nrow(X_out)
nu_vec = c(1:100)
delta = 1e-2
Psi_p = (Np * Rp + delta * diag(R0)) / (Np + delta)
llhs = log_F(R0, Psi_p, N, nu_vec)
plot(nu_vec, llhs, xlab = paste0("nu value; setting delta = ", delta), ylab = "log-likelihood of matrix F-distribution")
print(paste0("Optimal nu is ", nu_vec[which.max(llhs)]))[1] "Optimal nu is 74"When \(\delta = 0.01\) we have larger \(\nu\), which is interesting. Further increasing \(\delta\) does not lead to a bigger optimal \(\nu\).
N = nrow(X_in)
Np = nrow(X_out)
nu_vec = c(1:100)
delta = 1e-1
Psi_p = (Np * Rp + delta * diag(R0)) / (Np + delta)
llhs = log_F(R0, Psi_p, N, nu_vec)
plot(nu_vec, llhs, xlab = paste0("nu value; setting delta = ", delta), ylab = "log-likelihood of matrix F-distribution")
print(paste0("Optimal nu is ", nu_vec[which.max(llhs)]))[1] "Optimal nu is 29"Now let’s try to add \(\text{diag}(R')\) instead of \(\text{diag}(R_0)\). The optimal \(\nu\) is not as large. So it seems like setting the prior as \(\text{diag}(R_0)\) is better.
N = nrow(X_in)
Np = nrow(X_out)
nu_vec = c(1:100)
delta = 1e-2
Psi_p = (Np * Rp + delta * diag(Rp)) / (Np + delta)
llhs = log_F(R0, Psi_p, N, nu_vec)
plot(nu_vec, llhs, xlab = paste0("nu value; setting delta = ", delta), ylab = "log-likelihood of matrix F-distribution")
print(paste0("Optimal nu is ", nu_vec[which.max(llhs)]))[1] "Optimal nu is 31"Model 5
\[\mathrm{IW}(R_0 | \nu \Psi', \nu + p + 1)\]
delta = 1e-6
Psi_p = (Np * Rp + delta * diag(Rp)) / (Np + delta)
llhs = log_iw(R0, Psi_p, nu_vec)
plot(nu_vec, llhs, xlab = paste0("nu value; setting delta = ", delta), ylab = "log-likelihood of IW")
print(paste0("Optimal nu is ", nu_vec[which.max(llhs)]))[1] "Optimal nu is 1"delta = 1e-2
Psi_p = (Np * Rp + delta * diag(Rp)) / (Np + delta)
llhs = log_iw(R0, Psi_p, nu_vec)
plot(nu_vec, llhs, xlab = paste0("nu value; setting delta = ", delta), ylab = "log-likelihood of IW")
print(paste0("Optimal nu is ", nu_vec[which.max(llhs)]))[1] "Optimal nu is 1"They are both bad, because in the pdf of IW, we take the inverse of \(R_0\) not \(R'\) (out-of-sample cov. mat.). So adding a small regularization to \(R'\) does not help and the PDF is small due to the singularity of \(R_0\). I can not make \(\nu\) any larger with Inverse-Wishart formulation here. But recall that in the Variational Inference (jointly with SuSiE), the estimated \(R_0\) does borrow eigenvectors of \(R'\) so that the estimated \(\nu\) might be larger.
In conclusion, jointly learning \(\delta\) can help to achieve larger \(\nu\), hence borrowing more strength from the out-of-sample LD (if appropriately adjusted by \(\delta\)).
==============
Extra: We try the model
\[\mathrm{F}(R_0 | \frac{\nu \Psi''}{N}, N, \nu + 2)\] where \(\Psi'' = \dfrac{N' \text{diag}(R_0)^{1/2} \text{diag}(R')^{-1/2} R' \text{diag}(R')^{-1/2} \text{diag}(R_0)^{1/2} + \delta \text{diag}(R_0)}{N' + \delta}\). But the estimated optimal \(\nu\) seems not as large.
N = nrow(X_in)
Np = nrow(X_out)
nu_vec = c(1:100)
delta = 1e-2
Psi_pp = (Np * diag(diag(R0)^(0.5)) %*% cov2cor(Rp) %*% diag(diag(R0)^(0.5)) + delta * diag(R0)) / (Np + delta)
llhs = log_F(R0, Psi_p, N, nu_vec)
plot(nu_vec, llhs, xlab = paste0("nu value when delta = ", delta), ylab = "log-likelihood of matrix F-distribution")
print(paste0("Optimal nu is ", nu_vec[which.max(llhs)]))[1] "Optimal nu is 31"
sessionInfo()R version 4.3.2 (2023-10-31)
Platform: aarch64-apple-darwin20 (64-bit)
Running under: macOS 26.1
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_2.3.10
[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_0.52 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