>I've been doing research on extracting mutual information from noisy, shifted copies of a ground truth signal, and it turns out there is a crossover point where recovering the ground truth essentially becomes impossible (in an information theoretic sense). What's interesting is that this crossover point is sharp and it looks a lot like a phase transition that one might see in statistical physics.

That's awesome! Can you link me to your research? I've been writing a similar paper on the information theory behind probabilistic programming and (possibly) deep learning.

>We need more research that provides limits on what we are capable of predicting from a given dataset — an upper bound, in other words. This would let us know if it's worth it to spend time trying to get more predictivity out of a dataset, or if the data just simply doesn't contain sufficient enough information.

Definitely! Not only is the Shannon entropy of large or high-dimensional datasets computationally obscene to estimate nonparametrically, what we actually care about is the conditional entropy given an untrained model, and the degree to which we can reduce it by training the model.