Is the Hoeffding Inequality really valid for each bin despite nonrandom sampling?
The multiple bin analogy of picking the best h is a very helpful way of visualizing the situation, and I totally get how the union bound sets an upper limit on the probability of exceeding the error threshold. What I am actually questioning is whether those individual probabilities that compose the union bound are correct. I can see that they are just the individual Hoeffding Inequalities for each h, but is the Hoeffding Inequality really valid for all those h's in spite of the fact that we are NOT taking random samples from each "bin"? We are only picking our marbles (x's) ONCE, and then repicking the same marbles from each bin (yes, the redgreen colors of those marbles can change, based on the specific h, but aren't they the same marbles (x's)?).
Thanks for your help!
