Bayesian Single Effect Regression model

Last updated: 2018-09-05

• ✔ R Markdown file: up-to-date

  Great! Since the R Markdown file has been committed to the Git repository, you know the exact version of the code that produced these results.

• ✔ Environment: empty

  Great job! The global environment was empty. Objects defined in the global environment can affect the analysis in your R Markdown file in unknown ways. For reproduciblity it’s best to always run the code in an empty environment.

• ✔ Seed: set.seed(20180529)

  The command set.seed(20180529) was run prior to running the code in the R Markdown file. Setting a seed ensures that any results that rely on randomness, e.g. subsampling or permutations, are reproducible.

• ✔ Session information: recorded

  Great job! Recording the operating system, R version, and package versions is critical for reproducibility.

• ✔ Repository version: dd57e1f

  Great! You are using Git for version control. Tracking code development and connecting the code version to the results is critical for reproducibility. The version displayed above was the version of the Git repository at the time these results were generated.
  
  Note that you need to be careful to ensure that all relevant files for the analysis have been committed to Git prior to generating the results (you can use wflow_publish or wflow_git_commit). workflowr only checks the R Markdown file, but you know if there are other scripts or data files that it depends on. Below is the status of the Git repository when the results were generated:

  
  Ignored files:
      Ignored:    .Rhistory
      Ignored:    .Rproj.user/
      Ignored:    analysis/.Rhistory
      Ignored:    docs/.DS_Store
      Ignored:    docs/figure/
  
  Unstaged changes:
      Modified:   analysis/_site.yml
  
  

  Note that any generated files, e.g. HTML, png, CSS, etc., are not included in this status report because it is ok for generated content to have uncommitted changes.

Building on this simple univariate regression, we define the ‘single effect (multiple) regression’ model (SER model), to be a multiple regression model in which it is assumed that exactly one of p covariates has a non-zero effect.

The SER model is \[ \begin{align*} \mathbf{y} &= X \boldsymbol{\beta} + \boldsymbol{\epsilon} \quad \boldsymbol{\epsilon}\sim N_{n}(0, \frac{1}{\tau}I) \\ \boldsymbol{\beta} &= \boldsymbol{\gamma} \beta \\ \boldsymbol{\gamma} &\sim Multinomial(1,\boldsymbol{\pi}) \\ \beta &\sim N(0, \frac{1}{\tau_{0}}) \end{align*} \] As we have shown in Bayesian Linear Regression, the posterior on the indicator vector \(\boldsymbol{\gamma}\) is: \[ \boldsymbol{\gamma}|\mathbf{y},X,\tau,\tau_{0},\boldsymbol{\pi} \sim Multinomial(1, \boldsymbol{\alpha}) \\ \alpha_{j} = \frac{\pi_{j}BF(\mathbf{y}, \mathbf{x}_{j})}{\sum_{k}\pi_{k}BF(\mathbf{y}, \mathbf{x}_{k})} \] The \(\boldsymbol{\alpha}\) is the vector of ‘Posterior Inclusion Probabilitie’.

Conditional on the jth element of \(\boldsymbol{\gamma}\) being non-zero, \[ \beta | \gamma_{j} = 1, \mathbf{y}, X, \tau, \tau_{0} \sim N(\mu_{1j}, \frac{1}{\tau_{1j}}) \\ \tau_{1j} = \tau\mathbf{x}_{j}^{T}\mathbf{x}_{j} + \tau_{0} \\ \mu_{1j} = \frac{\tau\mathbf{x}_{j}^{T}\mathbf{y}}{\tau_{1j}} \]

The variational lower bound: \[ \log p(\mathbf{y}) = L(q(\boldsymbol{\beta})) + KL\left[q(\boldsymbol{\beta} \| p(\boldsymbol{\beta}|\mathbf{y})\right] \]

Here we have the exact posterior \(p(\boldsymbol{\beta}|\mathbf{y})\). So the lower bound is same as the log likelihood.

\[ \begin{align*} p(\mathbf{y}|X, \boldsymbol{\pi}, \tau_{0}, \tau) &= \int p(\mathbf{y}, \beta, \boldsymbol{\gamma}|X, \boldsymbol{\pi}, \tau_{0}, \tau) d \boldsymbol{\gamma} d \beta \\ &= \sum_{j=1}^{p}\pi_{j}\int \left(\frac{1}{\sqrt{2\pi/\tau}}\right)^{n}\exp\left(-\frac{\tau\|\mathbf{y}-X\boldsymbol{\gamma}\beta\|^{2}}{2}\right) N(\beta; 0, 1/\tau_{0}) d \beta \\ &= \sum_{j=1}^{p}\pi_{j} N(\hat{\beta}_{j}; 0, \frac{1}{\tau\mathbf{x}_{j}^{T}\mathbf{x}_{j}} + \frac{1}{\tau_{0}}) \\ &= \sum_{j=1}^{p}\pi_{j} \frac{N(\hat{\beta}_{j}; 0, \frac{1}{\tau\mathbf{x}_{j}^{T}\mathbf{x}_{j}} + \frac{1}{\tau_{0}})}{N(\mathbf{y}; 0, \frac{1}{\tau}I) } N(\mathbf{y}; 0, \frac{1}{\tau}I) \\ &= \sum_{j=1}^{p}\pi_{j} \frac{N(\hat{\beta}_{j}; 0, \frac{1}{\tau\mathbf{x}_{j}^{T}\mathbf{x}_{j}} + \frac{1}{\tau_{0}})}{N(\hat{\beta}_{j}; 0, \frac{1}{\tau\mathbf{x}_{j}^{T}\mathbf{x}_{j}})} N(\mathbf{y}; 0, \frac{1}{\tau}I) \\ &= \sum_{j=1}^{p} \pi_{j} BF(\mathbf{y}, \mathbf{x}_{j}) N(\mathbf{y}; 0, \frac{1}{\tau}I) \end{align*} \]

\[ \log p(\mathbf{y}) = L(q(\boldsymbol{\beta})) = \log \left[\sum_{j=1}^{p} \pi_{j} BF(\mathbf{y}, \mathbf{x}_{j})\right] + \log N(\mathbf{y}; 0, \frac{1}{\tau}I) \]

sessionInfo()

R version 3.5.1 (2018-07-02)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS High Sierra 10.13.6

Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/3.5/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.5/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     

loaded via a namespace (and not attached):
 [1] workflowr_1.1.1   Rcpp_0.12.18      digest_0.6.15    
 [4] rprojroot_1.3-2   R.methodsS3_1.7.1 backports_1.1.2  
 [7] git2r_0.23.0      magrittr_1.5      evaluate_0.11    
[10] stringi_1.2.4     whisker_0.3-2     R.oo_1.22.0      
[13] R.utils_2.6.0     rmarkdown_1.10    tools_3.5.1      
[16] stringr_1.3.1     yaml_2.2.0        compiler_3.5.1   
[19] htmltools_0.3.6   knitr_1.20       

This reproducible R Markdown analysis was created with workflowr 1.1.1