I read one of the papers. About the specific question you have: given a string of bits s, they’re making the choice to associate the empirical distribution to s, as if s was generated by an iid Bernoulli process. So if s has 10 zero bits and 30 one bits, its associated empirical distribution is Ber(3/4). This is the distribution which they’re calculating the entropy of. I have no idea on what basis they are making this choice.

The rest of the paper didn’t make sense to me - they are somehow assigning a number N of “information states” which can change over time as the memory cells fail. I honestly have no idea what it’s supposed to mean and kinda suspect the whole thing is rubbish.

Edit: after reading the author’s quotes from the associated hype article I’m 100% sure it’s rubbish. It’s also really funny that they didn’t manage to catch the COVID-19 research hype train so they’ve pivoted to the simulation hypothesis.

Mr. Costantino said the design was not at fault and that the towering mast, which stood 237 feet tall, had not created “any kind of problem.”

“The ship was an unsinkable ship,” he said. “I say it, I repeat it.”

- Designer of sunken ship