Re: Isn't the bin (your data set) the sample?
mu denotes probability of green in the entire space, outside of D included.
In a marble in a bag example, bin is the entire space and the N marbles you picked are your data set. i.e. you do not know the colors of the rest of the marbles in the bin.
Consider the following example. There are 10000 marbles in the bag and you want to know the proportion of red and black marbles (proportion of red: mu). You can figure out the exact ratio by taking out all the marbles and counting all of them. But say you have to figure this out in a limited time, and can only afford to look at 100 marbles. You counted 30 red marbles, and 70 black marbles. Then v is 0.3. You do not know if marbles outside of your dataset agrees with it. But Hoeffding Inequality provides a bound for the probability for values of mu based on v.
