We could remove information from For instance, could be a bit indicating whether the temperature is above 100°C
I don’t understand how this is less information than a bit indicating whether the temperature is above 50C. Specifically, given a bit telling you whether the temperature is above 50C, how do you know whether the temperature is above 100C or between 50C and 100C?
Let’s say every day at the office, we get three boxes of donuts, numbered 1, 2, and 3. I grab a donut from each box, plunk them down on napkins helpfully labeled X1, X2, and X3. The donuts vary in two aspects: size (big or small) and flavor (vanilla or chocolate). Across all boxes, the ratio of big to small donuts remains consistent. However, Boxes 1 and 2 share the same vanilla-to-chocolate ratio, which is different from that of Box 3.
Does the correlation between X1 and X2 imply that there is no natural latent? Is this the desired behavior of natural latents, despite the presence of the common size ratio? (and the commonality that I’ve only ever pulled out donuts; there has never been a tennis ball in any of the boxes!)
If so, why is this what we want from natural latents? If not, how does a natural latent arise despite the internal correlation?