How do I use FDR in this situation? Firstly, I am not a statistician so apologies if attempt of a solution is ridiculous.
I am trying to come up with an algorithm that estimates a vector $v$ thats contaminated with noise
Suppose we have vector $v$ (which has entries $1, \ldots, l$) of $0's$ and $1's$. I want this function to be independent of how $v$ is distributed so no inference of its distribution should be used, but for now lets assume its uniform (also each entry of the vector is independent)
Now I make $n$ copies of this vector and then add Gaussian noise with mean 0 and variance 1 to each copy (independently) lets call them $x_1, \ldots x_n$. I want to use $x_1 \ldots x_n$ to try and recover $v$
I want to try and estimate the original using false discovery rate but I'm not quite sure how I would apply it to this situation
Suggest I'm trying to estimate the $i$th entry of $v$. According to wikipedia (BH procedure) I would have a collection of null hypothesis, I'm a little unconfident in thinking they're
$H_1 =$ the $i$th entry is $0$ with added Gaussian noise of mean $0$
$H_2 = $ the $i$th entry is $1$ with added Gaussian noise of mean $0$
Then wikipedia (Benjamini–Hochberg procedure) says I should have some $p$-values, but now I'm a little lost, p-values corresponding to what data? My best guess is that they're
$p_1$ is the probability of getting data at least as extreme as $\{x_i(1), \ldots,x_i(l)\}$ under the assumption of $H_1$ 
$p_2$ is the probability of getting data at least as extreme as $\{x_i(1), \ldots,x_i(l)\}$ under the assumption of $H_2$
(I think I would use a t-test to calculate these p-values with dgrees of freedom $n-1$?) 
Now I am meant to order them, for simplicity lets assume they're already ordered
Lets set false discovery rate $\alpha = 0.05$. 
Suppose $k=1$ is the largest $k$ such that $P_k \leq \frac km \alpha = 0.025k$ and so then I reject $H_1$. Is it now a fair to say that my best guess at $v(i)$ is that it is $1$? 
What should I infer if all or none $k$ are such that $P_k \leq \frac km \alpha$
Does this look correct to you? 
I appreciate any feedback 
Thanks
 A: Here's my shot, it might be incorrect, so hopefully more people will chime in.
Given a $g\times1$ vector $v$ : {$1, ..., g$}  $\to$ {$1, 0$}, and an error vector $e$ ~ $Z(0,1)$, creating $n$ copies of $v+e$ results in a $g \times n$ matrix named $V$.
We want to recover $v$, knowing the distribution of $e$.  Each row $i$ of $V$ either has true $\mu = 0$ or $\mu = 1$, which we can assess by a simple $Z$ test for $\hat{\mu} = 0$ because we know the distribution of the noise.
We are interested solely in true differences larger than 0, so we decide to do a one-tailed Z test.  This makes:
$H_0:$ $\mu_i = 0$
$H_1:$ $\mu_i > 0$
Setting $\alpha = 0.05$, critical $z = 1.6449$ for a one tailed $Z$ test.

Computing a $Z$ test P value for each row $V_i$ gives us a $g \times 1$ vector of P values, of which we must correct for multiple testing through FDR.
Because we set $\alpha = 0.05$, the chance of truly accepting the null is 0.95 for one test, and 0.95$^n$ for $n$ uncorrelated tests.  Obviously this must be corrected for.  One method would be Bonferroni, another would be FDR.  Bonferonni correction controls family wise error rate, wheras FDR controls the false discovery rate.  
Ordering your P values (but obviously maintaining their index values) and allowing for false discovery rate = 0.05, you would declare a test a significant result if $P_i < \frac{i}{g} \alpha$. This would mean that the effect overcomes the multiplicity (with 5% error).  If you have no values with $P_i < \frac{i}{g} \alpha$, it means that none of your values have been succesfully recovered.
However, the original one tailed $Z$ test only has $\beta - 1 = 0.26$ at $\alpha = 0.05$, so only $\approx$ 26% of your true 1 values can be recovered.  If $g$ is small, then perhaps noise overwhelmed signal in all your cases. 
There are definitely better ways, but I hope this helped a little bit.
