<!DOCTYPE html>

<html xmlns="http://www.w3.org/1999/xhtml">

<head>

<meta charset="utf-8" />
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
<meta name="generator" content="pandoc" />


<meta name="author" content="Jason Willwerscheid" />


<title>Notes on Computing the FLASH Objective</title>

<script src="site_libs/jquery-1.11.3/jquery.min.js"></script>
<meta name="viewport" content="width=device-width, initial-scale=1" />
<link href="site_libs/bootstrap-3.3.5/css/cosmo.min.css" rel="stylesheet" />
<script src="site_libs/bootstrap-3.3.5/js/bootstrap.min.js"></script>
<script src="site_libs/bootstrap-3.3.5/shim/html5shiv.min.js"></script>
<script src="site_libs/bootstrap-3.3.5/shim/respond.min.js"></script>
<script src="site_libs/jqueryui-1.11.4/jquery-ui.min.js"></script>
<link href="site_libs/tocify-1.9.1/jquery.tocify.css" rel="stylesheet" />
<script src="site_libs/tocify-1.9.1/jquery.tocify.js"></script>
<script src="site_libs/navigation-1.1/tabsets.js"></script>
<link href="site_libs/highlightjs-9.12.0/textmate.css" rel="stylesheet" />
<script src="site_libs/highlightjs-9.12.0/highlight.js"></script>
<link href="site_libs/font-awesome-4.5.0/css/font-awesome.min.css" rel="stylesheet" />

<style type="text/css">code{white-space: pre;}</style>
<style type="text/css">
  pre:not([class]) {
    background-color: white;
  }
</style>
<script type="text/javascript">
if (window.hljs) {
  hljs.configure({languages: []});
  hljs.initHighlightingOnLoad();
  if (document.readyState && document.readyState === "complete") {
    window.setTimeout(function() { hljs.initHighlighting(); }, 0);
  }
}
</script>



<style type="text/css">
h1 {
  font-size: 34px;
}
h1.title {
  font-size: 38px;
}
h2 {
  font-size: 30px;
}
h3 {
  font-size: 24px;
}
h4 {
  font-size: 18px;
}
h5 {
  font-size: 16px;
}
h6 {
  font-size: 12px;
}
.table th:not([align]) {
  text-align: left;
}
</style>


</head>

<body>

<style type = "text/css">
.main-container {
  max-width: 940px;
  margin-left: auto;
  margin-right: auto;
}
code {
  color: inherit;
  background-color: rgba(0, 0, 0, 0.04);
}
img {
  max-width:100%;
  height: auto;
}
.tabbed-pane {
  padding-top: 12px;
}
button.code-folding-btn:focus {
  outline: none;
}
</style>


<style type="text/css">
/* padding for bootstrap navbar */
body {
  padding-top: 51px;
  padding-bottom: 40px;
}
/* offset scroll position for anchor links (for fixed navbar)  */
.section h1 {
  padding-top: 56px;
  margin-top: -56px;
}

.section h2 {
  padding-top: 56px;
  margin-top: -56px;
}
.section h3 {
  padding-top: 56px;
  margin-top: -56px;
}
.section h4 {
  padding-top: 56px;
  margin-top: -56px;
}
.section h5 {
  padding-top: 56px;
  margin-top: -56px;
}
.section h6 {
  padding-top: 56px;
  margin-top: -56px;
}
</style>

<script>
// manage active state of menu based on current page
$(document).ready(function () {
  // active menu anchor
  href = window.location.pathname
  href = href.substr(href.lastIndexOf('/') + 1)
  if (href === "")
    href = "index.html";
  var menuAnchor = $('a[href="' + href + '"]');

  // mark it active
  menuAnchor.parent().addClass('active');

  // if it's got a parent navbar menu mark it active as well
  menuAnchor.closest('li.dropdown').addClass('active');
});
</script>


<div class="container-fluid main-container">

<!-- tabsets -->
<script>
$(document).ready(function () {
  window.buildTabsets("TOC");
});
</script>

<!-- code folding -->




<script>
$(document).ready(function ()  {

    // move toc-ignore selectors from section div to header
    $('div.section.toc-ignore')
        .removeClass('toc-ignore')
        .children('h1,h2,h3,h4,h5').addClass('toc-ignore');

    // establish options
    var options = {
      selectors: "h1,h2,h3",
      theme: "bootstrap3",
      context: '.toc-content',
      hashGenerator: function (text) {
        return text.replace(/[.\\/?&!#<>]/g, '').replace(/\s/g, '_').toLowerCase();
      },
      ignoreSelector: ".toc-ignore",
      scrollTo: 0
    };
    options.showAndHide = true;
    options.smoothScroll = true;

    // tocify
    var toc = $("#TOC").tocify(options).data("toc-tocify");
});
</script>

<style type="text/css">

#TOC {
  margin: 25px 0px 20px 0px;
}
@media (max-width: 768px) {
#TOC {
  position: relative;
  width: 100%;
}
}


.toc-content {
  padding-left: 30px;
  padding-right: 40px;
}

div.main-container {
  max-width: 1200px;
}

div.tocify {
  width: 20%;
  max-width: 260px;
  max-height: 85%;
}

@media (min-width: 768px) and (max-width: 991px) {
  div.tocify {
    width: 25%;
  }
}

@media (max-width: 767px) {
  div.tocify {
    width: 100%;
    max-width: none;
  }
}

.tocify ul, .tocify li {
  line-height: 20px;
}

.tocify-subheader .tocify-item {
  font-size: 0.90em;
  padding-left: 25px;
  text-indent: 0;
}

.tocify .list-group-item {
  border-radius: 0px;
}


</style>

<!-- setup 3col/9col grid for toc_float and main content  -->
<div class="row-fluid">
<div class="col-xs-12 col-sm-4 col-md-3">
<div id="TOC" class="tocify">
</div>
</div>

<div class="toc-content col-xs-12 col-sm-8 col-md-9">




<div class="navbar navbar-default  navbar-fixed-top" role="navigation">
  <div class="container">
    <div class="navbar-header">
      <button type="button" class="navbar-toggle collapsed" data-toggle="collapse" data-target="#navbar">
        <span class="icon-bar"></span>
        <span class="icon-bar"></span>
        <span class="icon-bar"></span>
      </button>
      <a class="navbar-brand" href="index.html">FLASHvestigations</a>
    </div>
    <div id="navbar" class="navbar-collapse collapse">
      <ul class="nav navbar-nav">
        <li>
  <a href="index.html">Home</a>
</li>
<li>
  <a href="about.html">About</a>
</li>
<li>
  <a href="license.html">License</a>
</li>
      </ul>
      <ul class="nav navbar-nav navbar-right">
        <li>
  <a href="https://github.com/willwerscheid/FLASHvestigations">
    <span class="fa fa-github"></span>
     
  </a>
</li>
      </ul>
    </div><!--/.nav-collapse -->
  </div><!--/.container -->
</div><!--/.navbar -->

<!-- Add a small amount of space between sections. -->
<style type="text/css">
div.section {
  padding-top: 12px;
}
</style>

<div class="fluid-row" id="header">



<h1 class="title toc-ignore">Notes on Computing the FLASH Objective</h1>
<h4 class="author"><em>Jason Willwerscheid</em></h4>
<h4 class="date"><em>7/17/2018</em></h4>

</div>


<p><strong>Last updated:</strong> 2018-07-17</p>
<strong>workflowr checks:</strong> <small>(Click a bullet for more information)</small>
<ul>
<li>
<p><details> <summary> <strong style="color:blue;">✔</strong> <strong>R Markdown file:</strong> up-to-date </summary></p>
<p>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.</p>
</details>
</li>
<li>
<p><details> <summary> <strong style="color:blue;">✔</strong> <strong>Environment:</strong> empty </summary></p>
<p>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.</p>
</details>
</li>
<li>
<p><details> <summary> <strong style="color:blue;">✔</strong> <strong>Seed:</strong> <code>set.seed(20180714)</code> </summary></p>
<p>The command <code>set.seed(20180714)</code> 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.</p>
</details>
</li>
<li>
<p><details> <summary> <strong style="color:blue;">✔</strong> <strong>Session information:</strong> recorded </summary></p>
<p>Great job! Recording the operating system, R version, and package versions is critical for reproducibility.</p>
</details>
</li>
<li>
<p><details> <summary> <strong style="color:blue;">✔</strong> <strong>Repository version:</strong> <a href="https://github.com/willwerscheid/FLASHvestigations/tree/e811943688be6d783d32c4d45af52d5f09fe1e05" target="_blank">e811943</a> </summary></p>
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. <br><br> 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 <code>wflow_publish</code> or <code>wflow_git_commit</code>). 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:
<pre><code>
Ignored files:
    Ignored:    .DS_Store
    Ignored:    .Rhistory
    Ignored:    .Rproj.user/
    Ignored:    docs/.DS_Store
    Ignored:    docs/figure/.DS_Store

</code></pre>
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. </details>
</li>
</ul>
<details> <summary> <small><strong>Expand here to see past versions:</strong></small> </summary>
<ul>
<table style="border-collapse:separate; border-spacing:5px;">
<thead>
<tr>
<th style="text-align:left;">
File
</th>
<th style="text-align:left;">
Version
</th>
<th style="text-align:left;">
Author
</th>
<th style="text-align:left;">
Date
</th>
<th style="text-align:left;">
Message
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
Rmd
</td>
<td style="text-align:left;">
<a href="https://github.com/willwerscheid/FLASHvestigations/blob/e811943688be6d783d32c4d45af52d5f09fe1e05/analysis/obj_notes.Rmd" target="_blank">e811943</a>
</td>
<td style="text-align:left;">
Jason Willwerscheid
</td>
<td style="text-align:left;">
2018-07-17
</td>
<td style="text-align:left;">
wflow_publish(“analysis/obj_notes.Rmd”)
</td>
</tr>
<tr>
<td style="text-align:left;">
html
</td>
<td style="text-align:left;">
<a href="https://cdn.rawgit.com/willwerscheid/FLASHvestigations/06211851c6ed5c7c2e1870b1d8ab53d50186e55d/docs/obj_notes.html" target="_blank">0621185</a>
</td>
<td style="text-align:left;">
Jason Willwerscheid
</td>
<td style="text-align:left;">
2018-07-17
</td>
<td style="text-align:left;">
Build site.
</td>
</tr>
<tr>
<td style="text-align:left;">
Rmd
</td>
<td style="text-align:left;">
<a href="https://github.com/willwerscheid/FLASHvestigations/blob/a0bccf91e8bdf3e50e532b290c9ae313c245a62c/analysis/obj_notes.Rmd" target="_blank">a0bccf9</a>
</td>
<td style="text-align:left;">
Jason Willwerscheid
</td>
<td style="text-align:left;">
2018-07-17
</td>
<td style="text-align:left;">
wflow_publish(“analysis/obj_notes.Rmd”)
</td>
</tr>
</tbody>
</table>
</ul>
<p></details></p>
<hr />
<div id="indirect-method" class="section level2">
<h2>Indirect method</h2>
<p>Recall the FLASH model: <span class="math display">\[ Y = LF&#39; + E \]</span></p>
<p>When updating loading <span class="math inline">\(l_k\)</span>, we are optimizing over <span class="math inline">\(g_{l_k}\)</span> and <span class="math inline">\(q_{l_k}\)</span>. <span class="math inline">\(g_{l_k} \in \mathcal{G}\)</span> is the prior on the elements of the <span class="math inline">\(k\)</span>th column of the loadings matrix: <span class="math display">\[ l_{1k}, \ldots, l_{nk} \sim^{iid} g_{l_k} \]</span> <span class="math inline">\(q_{l_k}\)</span> is an arbitrary distribution which enters the problem via the variational approach. For convenience, I drop the subscripts in the following.</p>
<p>The part of the objective that depends on <span class="math inline">\(g\)</span> and <span class="math inline">\(q\)</span> is <span class="math display">\[ F(g, q) := E_q \left[ -\frac{1}{2} \sum_i (A_i l_i^2 - 2 B_i l_i) \right]
+ E_q \log \frac{g(\mathbf{l})}{q(\mathbf{l})} \]</span> with <span class="math display">\[ A_i = \sum_j \tau_{ij} Ef^2_j \text{ and }
B_i = \sum_j \tau_{ij} R_{ij} Ef_j, \]</span> (<span class="math inline">\(R\)</span> is the matrix of residuals (excluding factor <span class="math inline">\(k\)</span>) and <span class="math inline">\(EF_j\)</span> and <span class="math inline">\(EF^2_j\)</span> are the expected values of <span class="math inline">\(f_jk\)</span> and <span class="math inline">\(f_jk^2\)</span> with respect to the distribution <span class="math inline">\(q_{f_k}\)</span> fitted during the factor update.)</p>
<p>As Lemma 2 in the paper shows (see Appendix A.2), this expression is optimized by setting <span class="math inline">\(s_j^2 = A_j\)</span> and <span class="math inline">\(x_j = B_j s_j^2\)</span>, and then solving the EBNM problem, where the EBNM model is: <span class="math display">\[ \mathbf{x} = \mathbf{\theta} + \mathbf{e},\ \theta_1, \ldots, \theta_n \sim^{iid} g \]</span></p>
<p>Solving the EBNM problem gives <span class="math display">\[\hat{g} = {\arg \max}_g\ p(x \mid g) \]</span> and <span class="math display">\[ \hat{q} = p(\theta \mid x, \hat{g}) \]</span></p>
<p>Finally, to update the overall objective, we need to compute <span class="math inline">\(E_q \log \frac{g(\mathbf{l})}{q(\mathbf{l})}\)</span>. FLASH uses a clever trick, noticing that <span class="math display">\[ E_{\hat{q}} \log \frac{\hat{g}(\mathbf{l})}{\hat{q}(\mathbf{l})}
= F(\hat{g}, \hat{q}) + \frac{1}{2} 
E_{\hat{q}} \left[ \sum_j \log 2\pi s_j^2 + (1/s+j^2)(x_j - \theta_j)^2 \right]\]</span> (See Appendix A.4.)</p>
</div>
<div id="direct-method" class="section level2">
<h2>Direct method</h2>
<p>When using <code>ebnm_pn</code>, however, it seems possible to compute <span class="math inline">\(E_q \log \frac{g(\mathbf{l})}{q(\mathbf{l})}\)</span> directly. Since the elements <span class="math inline">\(l_1, \ldots, l_n\)</span> are i.i.d. from <span class="math inline">\(g\)</span> (by the FLASH model) and the posterior distributions are mutually independent (by the EBNM model), <span class="math display">\[ E_q \log \frac{g(\mathbf{l})}{q(\mathbf{l})}
= \sum_j E_{q_j} \log \frac{g(l_j)}{q(l_j)} \]</span></p>
<p>I drop the subscripts <span class="math inline">\(j\)</span>. Write <span class="math display">\[ g \sim \pi_0 \delta_0 + (1 - \pi_0) N(0, 1/a) \]</span> and <span class="math display">\[ q \sim \tilde{\pi}_0 \delta_0 + (1 - \tilde{\pi}_0) N(\tilde{\mu}, \tilde{\sigma}^2) \]</span> (I parametrize the normals differently to follow the code more closely.)</p>
<p>Then <span class="math display">\[\begin{aligned}
E_q \log \frac{g(l)}{q(l)}
&amp;= \tilde{\pi_0} \log \frac{\pi_0}{\tilde{\pi}_0}
+ \int (1 - \tilde{\pi}_0) \text{dnorm}(x; \tilde{\mu}, \tilde{\sigma}^2)
\log \frac{(1 - \pi_0)\text{dnorm}(x; 0, 1/a)}
{(1 - \tilde{\pi}_0)\text{dnorm}(x; \tilde{\mu}, \tilde{\sigma}^2)}\ dx \\
&amp;= \tilde{\pi_0} \log \frac{\pi_0}{\tilde{\pi}_0}
+ (1 - \tilde{\pi_0}) \log \frac{1 - \pi_0}{1 - \tilde{\pi}_0} \\
&amp;\ 
+ \int (1 - \tilde{\pi}_0) \text{dnorm}(x; \tilde{\mu}, \tilde{\sigma}^2)
\log \left( \sqrt{a \tilde{\sigma}^2} 
\exp \left( -\frac{ax^2}{2} + \frac{(x - \tilde{\mu})^2}{2 \tilde{\sigma}^2} \right) \right)
\ dx \\
&amp;= \tilde{\pi_0} \log \frac{\pi_0}{\tilde{\pi}_0}
+ (1 - \tilde{\pi_0}) \log \frac{1 - \pi_0}{1 - \tilde{\pi}_0} 
+ \frac{1 - \tilde{\pi_0}}{2} \log (a \tilde{\sigma}^2) \\
&amp;\ 
- \frac{(1 - \tilde{\pi}_0)a}{2} E_{N(x; \tilde{\mu}, \tilde{\sigma}^2)} x^2 + \frac{1 - \tilde{\pi}_0}{2 \tilde{\sigma}^2} E_{N(x; \tilde{\mu}, \tilde{\sigma}^2)} (x - \tilde{\mu})^2 \\
&amp;= \tilde{\pi_0} \log \frac{\pi_0}{\tilde{\pi}_0}
+ (1 - \tilde{\pi_0}) \log \frac{1 - \pi_0}{1 - \tilde{\pi}_0} 
+ \frac{1 - \tilde{\pi_0}}{2} \log (a \tilde{\sigma}^2)
- \frac{(1 - \tilde{\pi}_0)a}{2} (\tilde{\mu}^2 + \tilde{\sigma}^2)
+ \frac{1 - \tilde{\pi}_0}{2}
\end{aligned}\]</span></p>
<p>So we should be able to calculate <span class="math inline">\(E_q \log \frac{g(l)}{q(l)}\)</span> as follows:</p>
<pre class="r"><code>calc_KL &lt;- function(x, s, g) {
  pi0 &lt;- g$pi0
  w &lt;- 1 - g$pi0
  a &lt;- g$a
  
  wpost &lt;- ebnm:::wpost_normal(x, s, w, a) # 1 - \tilde{\pi}_0
  pi0post &lt;- 1 - wpost[wpost &lt; 1] # \tilde{\pi}_0
  pmean_cond &lt;- ebnm:::pmean_cond_normal(x, s, a) # \tilde{\mu}
  pvar_cond &lt;- ebnm:::pvar_cond_normal(s, a) # \tilde{\sigma}^2

  KLa &lt;- pi0post * log(pi0 / pi0post)
  KLb &lt;- wpost * log(w / wpost)
  KLc &lt;- (wpost / 2) * log(a * pvar_cond)
  KLd &lt;- -(wpost / 2) * a * (pvar_cond + pmean_cond^2)
  KLe &lt;- (wpost / 2)
  sum(KLa) + sum(KLb) + sum(KLc) + sum(KLd) + sum(KLe)
}</code></pre>
</div>
<div id="session-information" class="section level2">
<h2>Session information</h2>
<pre class="r"><code>sessionInfo()</code></pre>
<pre><code>R version 3.4.3 (2017-11-30)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS Sierra 10.12.6

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     

loaded via a namespace (and not attached):
 [1] workflowr_1.0.1   Rcpp_0.12.17      digest_0.6.15    
 [4] rprojroot_1.3-2   R.methodsS3_1.7.1 backports_1.1.2  
 [7] git2r_0.21.0      magrittr_1.5      evaluate_0.10.1  
[10] stringi_1.1.6     whisker_0.3-2     R.oo_1.21.0      
[13] R.utils_2.6.0     rmarkdown_1.8     tools_3.4.3      
[16] stringr_1.3.0     yaml_2.1.17       compiler_3.4.3   
[19] htmltools_0.3.6   knitr_1.20       </code></pre>
</div>

<!-- Adjust MathJax settings so that all math formulae are shown using
TeX fonts only; see
http://docs.mathjax.org/en/latest/configuration.html.  This will make
the presentation more consistent at the cost of the webpage sometimes
taking slightly longer to load. Note that this only works because the
footer is added to webpages before the MathJax javascript. -->
<script type="text/x-mathjax-config">
  MathJax.Hub.Config({
    "HTML-CSS": { availableFonts: ["TeX"] }
  });
</script>

<hr>
<p>
  This reproducible <a href="http://rmarkdown.rstudio.com">R Markdown</a>
  analysis was created with
  <a href="https://github.com/jdblischak/workflowr">workflowr</a> 1.0.1
</p>
<hr>


</div>
</div>

</div>

<script>

// add bootstrap table styles to pandoc tables
function bootstrapStylePandocTables() {
  $('tr.header').parent('thead').parent('table').addClass('table table-condensed');
}
$(document).ready(function () {
  bootstrapStylePandocTables();
});


</script>

<!-- dynamically load mathjax for compatibility with self-contained -->
<script>
  (function () {
    var script = document.createElement("script");
    script.type = "text/javascript";
    script.src  = "https://mathjax.rstudio.com/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML";
    document.getElementsByTagName("head")[0].appendChild(script);
  })();
</script>

</body>
</html>