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<h1 class="title toc-ignore">Trying method of moments for estimating prior in EB</h1>
<h4 class="author"><em>Matthew Stephens</em></h4>
<h4 class="date"><em>2017-12-09</em></h4>
</div>
<!-- The file analysis/chunks.R contains chunks that define default settings
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<p><strong>Last updated:</strong> 2017-12-12</p>
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package git2r is installed -->
<p><strong>Code version:</strong> ce26fc9</p>
<div id="idea" class="section level1">
<h1>Idea</h1>
<p>The idea is to investigate method of moments estimation for very fast EB procedures like EBayesThresh.</p>
<p>Suppose <span class="math inline">\(x_j = b_j + N(0,s_j^2)\)</span> and <span class="math inline">\(b_j \sim \pi_0 \delta_0 + (1-\pi_0) DExp(a)\)</span> where <span class="math inline">\(Dexp\)</span> means the double exponential (Laplace) with rate <span class="math inline">\(a\)</span>.</p>
<p>Then the even moments of <span class="math inline">\(b_j\)</span> are <span class="math inline">\(E(b_j^n) = (1-\pi_0) n! (1/a)^n\)</span>.</p>
<p>Also, 2nd and 4th moments of <span class="math inline">\(N(0,s_j^2)\)</span> are <span class="math inline">\(s_j^2\)</span> and <span class="math inline">\(3s_j^4\)</span>.</p>
<p>Thus, second and 4th moments of <span class="math inline">\(x_j\)</span> are: <span class="math display">\[E(x_j^2) = s_j^2 + 2(1-\pi_0)/a^2\]</span></p>
<p><span class="math display">\[E(x_j^4) = 3s_j^4 + 24(1-\pi_0)a^4 + 6 s_j^2 [2(1-\pi_0)a^2] \]</span>.</p>
<p>Let <span class="math inline">\(m_2 := E(x_j^2 - s_j^2)\)</span> and <span class="math inline">\(m_4:= E(x_j^4 - 3s_j^4)\)</span>. Then</p>
<p><span class="math display">\[E(m_2) = 2(1-\pi_0)/a^2\]</span>. <span class="math display">\[E(m_4) = 24[(1-\pi_0)/a^2] + 12s_j^2 (1-\pi_0)/a^2\]</span> <span class="math display">\[= m_2 (12/a^2 + 6s_j^2)\]</span> So we can solve to give <span class="math display">\[a^2 = 12/ (m_4/m_2 - 6s_j^2)\]</span></p>
<p>OK, so I’ve been sloppy with subscripts on the <span class="math inline">\(m_4\)</span> and <span class="math inline">\(m_2\)</span>… need to sort that out.</p>
<p>We we will try it with constant <span class="math inline">\(s_j=1\)</span>.</p>
</div>
<div id="simulate-some-data" class="section level1">
<h1>Simulate some data</h1>
<pre class="r"><code> eb_mm = function(x,s){
m4 = mean(x^4-3*s^4)
m2 = mean(x^2-s^2)
a2 = 12/((m4/m2) - 6*mean(s^2))
pi0 = 1- a2*m2/2
if(a2<0){a2=0; pi0=1}
if(pi0<0){pi0=0; a2 =2/m2}
if(pi0>1){pi0=1; a2 = 1}
return(list(pi0=pi0,a=sqrt(a2)))
}
set.seed(1)
n=100000
e = rnorm(n)
b = rexp(n,rate=2)
x = b+e
eb_mm(x,1)</code></pre>
<pre><code>$pi0
[1] 0
$a
[1] 2.003444</code></pre>
<pre class="r"><code> EbayesThresh::wandafromx(x,1,FALSE)</code></pre>
<pre><code>$w
[1] 1
$a
[1] 2.00342</code></pre>
<pre class="r"><code> b2 =b
b2[1:(n/2)] = 0
x2 = b2 + e
eb_mm(x2,1)</code></pre>
<pre><code>$pi0
[1] 0.4384081
$a
[1] 2.086427</code></pre>
<pre class="r"><code> EbayesThresh::wandafromx(x2,1,FALSE)</code></pre>
<pre><code>$w
[1] 0.5581188
$a
[1] 2.079653</code></pre>
<pre class="r"><code> EbayesThresh:::wandafromx.mle(x2,1)</code></pre>
<pre><code>$w
[1] 0.558119
$a
[1] 2.079654</code></pre>
<p>Try a case that is “nearly null”. Note that here the original Ebayesthresh approach based on the beta function is less accurate, presumably due to numeric issues.</p>
<pre class="r"><code> b3 =b
b3[1:(0.99*n)] = 0
x3 = b3 + e
eb_mm(x3,1)</code></pre>
<pre><code>$pi0
[1] 0.9088107
$a
[1] 3.961431</code></pre>
<pre class="r"><code> EbayesThresh::wandafromx(x3,1,FALSE)</code></pre>
<pre><code>$w
[1] 1
$a
[1] 13.08551</code></pre>
<pre class="r"><code> EbayesThresh:::wandafromx.mle(x3,1)</code></pre>
<pre><code>$w
[1] 0.09129917
$a
[1] 3.961397</code></pre>
</div>
<div id="thoughts" class="section level1">
<h1>Thoughts</h1>
<ul>
<li><p>hard part is when pi0 is very close to 1 but not 1. That might be worth thinking about. Maybe some upper quantiles would be better than 4th moment?</p></li>
<li><p>the case where s is not constant - probably want to work with E(x/s) rather than E(x) ?</p></li>
</ul>
<div id="session-information" class="section level2">
<h2>Session information</h2>
<!-- Insert the session information into the document -->
<pre class="r"><code>sessionInfo()</code></pre>
<pre><code>R version 3.3.2 (2016-10-31)
Platform: x86_64-apple-darwin13.4.0 (64-bit)
Running under: OS X El Capitan 10.11.6
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] MASS_7.3-47 backports_1.1.1 magrittr_1.5
[4] rprojroot_1.2 tools_3.3.2 htmltools_0.3.6
[7] yaml_2.1.14 Rcpp_0.12.14 stringi_1.1.5
[10] rmarkdown_1.7 EbayesThresh_1.5-0 knitr_1.17
[13] wavethresh_4.6.8 git2r_0.19.0 stringr_1.2.0
[16] digest_0.6.12 evaluate_0.10.1 </code></pre>
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