Sampling: If one sample produces multiple outputs, how many samples do I actually have to work with? A strangely worded question so let me explain. Consider the following image

I have a distribution $P$ at the top from which I can generate a number $x$. I have two sets of vectors, $S_1$ and $S_2$. Finally, I have a decider $D$ or classifier. Let's assume that $S_1$ has 10 vectors and $S_2$ has 5 vectors. A generated number $x$ from $P$ modifies vectors in both $S_1$ and $S_2$ (say multiplies them or whatever). Each modified vector then produces a 0 or 1 after it goes through the decider. My goal is to fit a distribution to the decider, and it is important for me to know how many samples I have so I can say what my confidence intervals are.   
Here's the question: If I generate one $x$ from $P$ and use it to modify all the vectors in $S_1$ and $S_2$ and go down the chain, I get 15 outputs from the decider. So do I have 1 sample or do I have 15 samples? My guess is 1 but I can't explain why.
Now let's assume I generate $x_{1}$ from $P$ and use it to modify the vectors in $S_{1}$, and generate another $x_{2}$ from $P$ and use it to modify the vectors in $S_{2}$. Again, I have 15 outputs from the decider. So do I have 2 samples or 15? 
 A: You have one sample of $x$ and 15 sample vectors. 
I think you might be worried that your 15 sample vectors can be thought of as a $15 k$-dimensional representation of a one-dimensional manifold (where $k$ is the dimension of each vector). But if your goal is to best approximate the mapping from the space of vectors to the output of $D$ then you have a discriminative problem, which means you don't care about the distribution from which your input was obtained. You only care that your input samples decently span the domain of $D$. Whether your points decently span the domain if they're only transformed from a single sample $x$ depends on the transformation process.
Consider the following example. Suppose $f(x) = \mathcal{N}(x; 0,1)$ is a mean-zero unit Gaussian distribution. You are given a data set of $N$ input-output pairs $\{x_i, y_i\}$, where the $x$ values are drawn from a different distribution, namely $g(x) = \mathcal{N}(x; 3, 25)$, and $y$ is evaluated as $y = f(x) + \epsilon$ where $\epsilon$ is a zero-mean Gaussian noise term (for example, a measurement error). Then, if you are trying to do a regression problem of finding $f(x)$ by evaluating $y$ then this would be a perfectly fine data set to do so, because outputs from $g(x)$ span the domain of $f(x)$ very well, even if $g(x)$ is nothing like $f(x).$ On the other hand, if $g(x)$ was simply a dirac delta function, then you would have a problem. 
You just have to ask yourself if the resulting vectors span the domain of $D$ well enough to let your estimation of $D$ properly generalize. Given what you told us in your question, we don't know if that's true. For example, it's possible that there are only $15$ possible realizations of a vector in the vector space with which you're concerned, and they're all obtained by generating any sample $x$ (the only thing $x$ determines is which vector gets represented by which transform). In that case, you only need one sample $x$ to perfectly describe $D.$ On the other hand, it could be that the value of every vector is just the value $x$ repeated $k$ times, in which case you get an extremely narrow piece of information from one sample (i.e. only one vector value). Obviously the true scenario will lie somewhere between those extreme cases.
