Posterior Inference with Gaussian Derivative Likelihood: Model and Result

GD-ASH Model

Recall the typical GD-ASH model is

\[ \begin{array}{l} \beta_j \sim \sum\pi_k N\left(0, \sigma_k^2\right) \ ;\\ \hat\beta_j = \beta_j + \hat s_j z_j \ ;\\ z_j \sim N\left(0, 1\right), \text{ correlated} \ . \end{array} \] Then we are fitting the empirical distribution of \(z\) with Gaussian derivatives

\[ f(z) = \sum w_l\frac{1}{\sqrt{l!}}\varphi^{(l)}(z) \ . \] Therefore, in essence, we are changing the likelihood of \(\hat\beta_j | \hat s_j, \beta_j\) from correlated \(N\left(\beta_j, \hat s_j^2\right)\) to independent \(\frac{1}{\hat s_j}f\left(\frac{\hat\beta_j - \beta_j}{\hat s_j}\right)\), which using Gaussian derivatives is

\[ \frac{1}{\hat s_j}\sum w_l \frac{1}{\sqrt{l!}} \varphi^{(l)}\left(\frac{\hat\beta_j - \beta_j}{\hat s_j}\right) \ . \] Note that when \(f = \varphi\) it becomes the independent \(N\left(\beta_j, \hat s_j^2\right)\) case.

GD-Lik Model

Therefore, if we use Gaussian derivatives instead of Gaussian as the likelihood, the posterior distribution of \(\beta_j | \hat s_j, \hat\beta_j\) should be

\[ \begin{array}{rcl} f\left(\beta_j \mid \hat s_j, \hat\beta_j\right) &=& \frac{ \displaystyle g\left(\beta_j\right) f\left(\hat\beta_j \mid \hat s_j, \beta_j \right) }{ \displaystyle\int g\left(\beta_j\right) f\left(\hat\beta_j \mid \hat s_j, \beta_j \right) d\beta_j }\\ &=& \frac{ \displaystyle \sum\pi_k\sum w_l \frac{1}{\sigma_k} \varphi\left(\frac{\beta_j - \mu_k}{\sigma_k}\right) \frac{1}{\hat s_j} \frac{1}{\sqrt{l!}} \varphi^{(l)}\left(\frac{\hat\beta_j - \beta_j}{\hat s_j}\right) }{ \displaystyle \sum\pi_k\sum w_l \int \frac{1}{\sigma_k} \varphi\left(\frac{\beta_j - \mu_k}{\sigma_k}\right) \frac{1}{\hat s_j} \frac{1}{\sqrt{l!}} \varphi^{(l)}\left(\frac{\hat\beta_j - \beta_j}{\hat s_j}\right) d\beta_j } \ . \end{array} \] The denominator readily has an analytic form which is

\[ \displaystyle \sum\pi_k \sum w_l \frac{\hat s_j^l}{\left(\sqrt{\sigma_k^2 + \hat s_j^2}\right)^{l+1}} \frac{1}{\sqrt{l!}} \varphi^{(l)}\left(\frac{\hat\beta_j - \mu_k}{\sqrt{\sigma_k^2 + \hat s_j^2}}\right) := \sum \pi_k \sum w_l f_{jkl} \ . \]

Posterior mean

After algebra, the posterior mean is given by

\[ E\left[\beta_j \mid \hat s_j, \hat \beta_j \right] = \int \beta_j f\left(\beta_j \mid \hat s_j, \hat\beta_j\right) d\beta_j = \displaystyle \frac{ \sum \pi_k \sum w_l m_{jkl} }{ \sum \pi_k \sum w_l f_{jkl} } \ , \] where \(f_{jkl}\) is defined as above and \[ m_{jkl} = - \frac{\hat s_j^l \sigma_k^2}{\left(\sqrt{\sigma_k^2 + \hat s_j^2}\right)^{l+2}} \frac{1}{\sqrt{l!}} \varphi^{(l+1)}\left(\frac{\hat\beta_j - \mu_k}{\sqrt{\sigma_k^2 + \hat s_j^2}}\right) + \frac{\hat s_j^l\mu_k}{\left(\sqrt{\sigma_k^2 + \hat s_j^2}\right)^{l+1}} \frac{1}{\sqrt{l!}} \varphi^{(l)}\left(\frac{\hat\beta_j - \mu_k}{\sqrt{\sigma_k^2 + \hat s_j^2}}\right) \ . \]

Local FDR

Assuming \(\mu_k \equiv 0\), the lfdr is given by \[ p\left(\beta_j = 0\mid \hat s_j, \hat \beta_j\right) = \frac{ \pi_0 \sum w_l \frac{1}{\hat s_j} \frac{1}{\sqrt{l!}} \varphi^{(l)}\left(\frac{\hat\beta_j}{\hat s_j}\right) }{ \sum \pi_k \sum w_l f_{jkl} } \ . \]

Local FSR

Right now the analytic form of lfsr using Gaussian derivatives is unavailable.

Simulation

The correlated \(N\left(0, 1\right)\) \(z\) scores are simulated from the GTEx/Liver data by the null pipeline. In order to get a better sense of the effectiveness of GD-ASH and GD-Lik, we are using data sets more distorted by correlation in the simulation. In particular, we are using an “inflation” batch, defined as the standard error of the correlated \(z\) no less than \(1.2\), and a “deflation” batch, defined as that no greater than \(0.8\). Out of \(1000\) simulated data sets, there are \(109\) inflation ones and \(99\) deflation ones.

In order to create realistic heterskedastic estimated standard error, \(\hat s_j\)’s are also simulated from the same null pipeline. Let \(\sigma^2 = \frac1n \sum\limits_{j = 1}^n \hat s_j^2\) be the average strength of the heteroskedastic noise, and the true effects \(\beta_j\)’s are simulated from \[ 0.6\delta_0 + 0.3N\left(0, \sigma^2\right) + 0.1N\left(0, \left(2\sigma\right)^2\right) \ . \]

Then let \(\hat\beta_j = \beta_j + \hat s_j z_j\). We are using \(\hat\beta_j\), \(\hat s_j\), along with \(\hat z_j = \hat\beta_j / \hat s_j\), \(\hat p_j= 2\left(1 - \Phi\left(\left|\hat z_j\right|\right)\right)\), as the summary statistics fed to GD-ASH and GD-Lik, as well as into BH, qvalue, locfdr, ASH for a comparison.

source("../code/gdash_lik.R")

z.mat = readRDS("../output/z_null_liver_777.rds")
se.mat = readRDS("../output/sebetahat_null_liver_777.rds")

z.sd = apply(z.mat, 1, sd)
inflation.index = (z.sd >= 1.2)
deflation.index = (z.sd <= 0.8)
z.inflation = z.mat[inflation.index, ]
se.inflation = se.mat[inflation.index, ]
## Number of inflation data sets
nrow(z.inflation)

[1] 109

z.deflation = z.mat[deflation.index, ]
se.deflation = se.mat[deflation.index, ]
## Number of deflation data sets
nrow(z.deflation)

[1] 99

Inflation data sets

Deflation data sets

Remarks

1. Would be nice to come up with a way to calculate lfsr in GD-Lik.
2. Many methods are too liberal for inflation cases and too conservative for deflation cases, showing a lack of robustness against correlation. Although, on average they probably seem about right.

Session information

sessionInfo()

R version 3.4.3 (2017-11-30)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS High Sierra 10.13.2

Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/3.4/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.4/Resources/lib/libRlapack.dylib

locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
[1] ashr_2.2-2        Rmosek_8.0.69     PolynomF_1.0-1    CVXR_0.94-4      
[5] REBayes_1.2       Matrix_1.2-12     SQUAREM_2017.10-1 EQL_1.0-0        
[9] ttutils_1.0-1    

loaded via a namespace (and not attached):
 [1] gmp_0.5-13.1      Rcpp_0.12.14      compiler_3.4.3   
 [4] git2r_0.20.0      R.methodsS3_1.7.1 R.utils_2.6.0    
 [7] iterators_1.0.9   tools_3.4.3       digest_0.6.13    
[10] bit_1.1-12        evaluate_0.10.1   lattice_0.20-35  
[13] foreach_1.4.4     yaml_2.1.16       parallel_3.4.3   
[16] Rmpfr_0.6-1       ECOSolveR_0.3-2   stringr_1.2.0    
[19] knitr_1.17        rprojroot_1.3-1   bit64_0.9-7      
[22] grid_3.4.3        R6_2.2.2          rmarkdown_1.8    
[25] magrittr_1.5      MASS_7.3-47       backports_1.1.2  
[28] codetools_0.2-15  htmltools_0.3.6   scs_1.1-1        
[31] stringi_1.1.6     doParallel_1.0.11 pscl_1.5.2       
[34] truncnorm_1.0-7   R.oo_1.21.0