It took me 10 years to understand entropy

You may be content with an interpretation restricted to talking about frequencies. I prefer a more general interpretation which can also - but not exclusively - refer to frequencies.

Even from a frequentist point of view I find perplexing your suggestion that nobody says that the probability of something is 100% when they are able to predict the outcome with certainty.

Probability may be a mathematical concept separate from reality but when it's applied to say things about the real world not all probability statements are equally good - just like the "moon made of cheese" model is not as good as any other even if it's mathematically flawless. This has nothing to do with Bayesian vs frequentist, by the way, empirical frequencies are not mathematical concepts separated from reality.

It's a perfectly frequentist thing to do to compare a sequence of probabilistic predictions to the realised outcomes to see how well-calibrated they are.

The astronomer that predicts P(total solar eclipse in 2023)=0%, P(t.s.e. 2024)=100%, P(t.s.e. 2025)=0%, P(t.s.e. 2026)=100%, etc. will score better than one who predicts P(t.s.e. 2023)=P(t.s.e. 2024)=P(t.s.e. 2025)=P(t.s.e. 2026)=2/3 or whatever is the long run frequency.

A weather forecaster that looks at satellite images will score better than one that predicts every day the average global rainfall. Being a frequentist doesn't prevent you from trying to do as well as you can.

I did already agree that you _can_ keep your model where the millenary-change ball may be either white or black and make calculations from it. (It just doesn't seem to me a good or useful description of that system once the precise state is known. You _can_ also change the model when you have more information and the updated model is objectively better. I think we will agree that from frequentist point of view predicting a white ball with 100% probability and getting it right every time is more accurate than a series of 50%/50% predictions. And the refined model can calculate the loooooong-term frequency of colours just as well.)

Yeah but it's a philosophical point of view. The bayesian sees this calibration process as the probability changing with more knowledge. The frequentist sees it as exactly you described a calibration... a correction on what was previously a more wrong probability.

>A weather forecaster that looks at satellite images will score better than one that predicts every day the average global rainfall. Being a frequentist doesn't prevent you from trying to do as well as you can.

Look at this way. Let's say I know that the next 100 days there will be sunny weather every day except for the 40th day, the 23rd day, the 12th day, the 16th day, the 67th day, the 21st day, the 19th day, the 98th day, and the 20th day. On those days there will be rain.

Is there any practicality if I say in the next 100 days there's a 9% chance of rain? I'd rather summarize that fact then regurgitate that mouthful. The statement and usage of the worse model is still relevant and has semantic meaning.

This is a example of deliberately choosing a model that is objectively worse then the previous one but it is chosen because it is more practical. In the same way we use approximate models, entropy is the same thing.

Personally I think this is irrelevant to the argument. I bring it up because associating the mathematical definitions of these concepts with practical daily intuition seems to be help your understanding.

I shouldn't have used the word "more wrong." A more accurate model is the better term. Similar to how general relativity is more accurate then classical mechanics.

>Sure. But the meaning no longer includes “as far as we know”

The definition of any model including entropy doesn't include as far as we know. And usage of any model less accurate or more accurate isn't exclusively based on that phrase.

Less accurate models are used all the time and they summarize details that the more accurate models fail to show. In fact all of analytics is based off aggregations which essentially lose detail and accuracy as data is processed. But the output of this data, however less fine grained summarizes an "average" that is otherwise invisible in the more detailed model.

Entropy is the same thing, it is a summary of the system. Greater knowledge of the system doesn't mean you discard the summary as useless.