Last updated: 2018-09-09

workflowr checks: (Click a bullet for more information)

  • ✔ 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.

  • ✔ Repository version: 610794c

    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:    .DS_Store
    Ignored:    .Rhistory
    Ignored:    .Rproj.user/
    Ignored:    analysis/.DS_Store
    Ignored:    data/LSI/
    Ignored:    docs/.DS_Store
    Ignored:    docs/figure/.DS_Store
    Ignored:    output/fig/

Unstaged changes:
    Modified:   analysis/smemo_2.rmd
    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.
Expand here to see past versions:
    File Version Author Date Message
    rmd 610794c Lei Sun 2018-09-09 wflow_publish(“converging_f.rmd”)

Introduction

Some theoretical justification.

A concrete example

Let’s start with a simple example. Let \((Z_1, Z_2, \ldots, Z_p)' \sim N_p(0, \Sigma)\), where \[\Sigma = \begin{bmatrix} 1 & 0.5 & \cdots & 0.5 \\ 0.5 & 1 & \cdots & 0.5 \\ \vdots & \vdots & \ddots & \vdots \\ 0.5 & 0.5 & \cdots & 1 \end{bmatrix} \ . \] In other words, \(Z_1, \ldots, Z_p\) are \(p\) correlated \(N(0, 1)\) random variables, with equal pairwise correlation of \(0.5\).

It’s easy to see that the limiting distribution of the empirical distribution of \(Z_1, \ldots, Z_p\) is \(f = N(\sqrt{1/2}Z, \sqrt{1/2}^2)\), where \(Z\) is a \(N(0, 1)\) random variable. \(f\) has some properties:

  1. \(f\) can be decomposed by Gaussian derivatives.

  2. \(f\) is a random normal distribution; the randomness comes from the fact that \(f\) has a random parameter, this case a random mean \(\sqrt{1/2}Z\).

  3. The distribution of this random mean is \(N(0, \sqrt{1/2}^2)\) whose shape and parameters are supposed to be determined by \(\Sigma\).

  4. The actual value of this random mean is determined by the realization of \(Z\), or eventually, the realization of \(Z_1, \ldots, Z_p\).

Generalization

In a generalization of the aforementioned example, we have \(Z_1, \ldots, Z_p\)

  • correlated \(N(0, 1)\),
  • every two \((Z_i, Z_j)\) bivariate normal with pairwise correlation \(\rho_{ij}\).

Let \(f\) be the limiting distribution of the empirical distribution of \(Z_1, \ldots, Z_p\). We should have

  1. \(Z_1, \ldots, Z_p\) are exchangeable.

  2. \(Z_1, \ldots, Z_p | W \overset{iid}{\sim} f(\cdot|W)\), where \(W := W_1, W_2, \ldots\) are the random parameters of \(f\).

  3. The joint distribution of \(W\) should be determined by \(\Sigma = [\rho_{ij}]_{p \times p}\), or at least we know that the first two moments of \(W\) are determined by \(\Sigma\).

  4. The actual values of \(W\) should be determined by the realization of \(Z_1, \ldots, Z_p\).

Can we justify this using de Finetti?

This reproducible R Markdown analysis was created with workflowr 1.1.1