Hoeffding Inequality
Hi,
On page 22, it says, "the hypothesis h is fixed before you generate the data set, and the probability is with respect to random data sets D; we emphasize that the assumption "h is fixed before you generate the data set" is critical to the validity of this bound". Few questions: 1. Does the "data set" in "generate the data set" refer to the marble (which is the data set D) we pick randomly from the jar? Or it refers to the set of outputs (red/green) of h(x) on D? 2. It keeps mentioning "h is fixed before you generate the data set". Does it mean in machine learning, a set of h should be predefined before seeing any training data and no h can be added to the set after seeing the training data? Thanks! 
Re: Hoeffding Inequality
Thanks for your quick reply Professor!
Now, I wonder why "we cannot just plug in g for h in the Hoeffding inequality". Given g is one of h's and for each h, Hoeffding inequality is valid for the upper bound of P[Ein(h)  Eout(h) > E]. Even g is picked after we look at all the outputs of all h's, g is still one of h's. So Hoeffding inequality should be still valid for g. No? Thanks! 
Re: Hoeffding Inequality
Quote:

Re: Hoeffding Inequality
Hi Professor,
I just have a hard time to understand that how choosing a hypothesis changes a theory  Hoeffding inequality. Let's say h1(x) < P1, h2(x) < P2. We choose h2 to be g. Then h2(x) < P2 is no longer true? I sort of understand your example because we pick the run of coin flipping that produces most heads, and so if we plot the graph, the graph indicates that Hoeffding inequality doesn't apply. But Hoeffding inequality is talking about probability and so the reality might be off a bit. Maybe I am in a wrong direction? :( 
Re: Hoeffding Inequality
It's a subtle point. There is "cherry picking" if we fish for a sample that has certain properties after many trials, instead of having a sample that is fairly drawn from a fixed hypothesis.
Statements involving probability are tricky because they don't guarantee a particular outcome, just the likelihood of getting that outcome. Therefore, changing the game to allow more trials or different conditions would change the probabilities. 
Re: Hoeffding Inequality
What the book states "e cannot just plug in g for h in the Hoeffding inequality" means that Hoeffding inequality is still true for g as it is one of the h's. But Hoeffding inequality seems failed for g because we are cherry picking.
Just like flipping an unbiased coin 1 million times, we should see 500K heads and 500K tails, but we might have "bad luck" that we see 1 million heads but the P(head) is still 0.5. Do I interpret correctly? Thanks a lot! 
Re: Hoeffding Inequality
Let me rephrase it. Let's say (like in Hoeffding) that a rare event has a probability of at most 1% of happening. If we make repeated independent trials looking for that event, each trial still gives a probability of at most 1% for that event to happen. Now, if we actively search for the case when that rare event actually happened among these many trials, we will succeed in finding it with probability much more than 1%.

Re: Hoeffding Inequality
I also have the same questions, and I read your replies
please consider if I have the correct conclusions: 1 we cannot plug "g" for "h" in inequality, because it depends on the sample we already selected, or in other words, we choose it deliberately (as the h with lowest error inside D) like selecting the bin which has the minimum frequency of heads. So! what if we select "g" randomly? (in a uniform distribution of hs ?) or to select a bin randomly, then can we use Hoeffding inequality for "g"? or still we should consider M, the H size? 2 which of the following interpretation for equation 1.6 are correct:
Second question: In "h is fixed before you generate the data set" I also can't understand your emphasis on "before". Do you want to say that h shouldn't change? because I feel h is independent from D then "before" or "after" doesn't mean much. We don't need to have an h in mind to be able to generate D, we can select D, then decide which h to use, then evaluate h over D, but we should use the same h for the test set, right? or maybe h is used somehow in generating D?! Anyway, I think you may mean it should be selected independently from D 
Re: Hoeffding Inequality
I think you can take a look on MaciekLeks' post for the experiment result of the Exercise 1.10 (in the book).
In my understanding: g is the final hypothesis that is known after the data set is generated (because the choice of final hypothesis is based on the specific data set). Before the data set is generated, all the information that we know about g is that g is one of the hypotheses in H (hence the M). h is a specific hypothesis that is an element of H, and I don't think that we are selecting h, I think we are selecting g instead. Quote:

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