back in 2021, when I still had space on my phone, I used to doodle on a sketches app and write down whatever came to mind. on january 4th it was this:
how am I to know if I am a big fish in a big pond? I am a fish with body dysmorphia and I haven't seen the full distribution of pond sizes.
it was a joke about comparing yourself to everyone around you. it's also a real statistics problem, and this is me finally taking it literally.
the fish has a fixed true length, L. it hops from pond to pond, and pond sizes are Pareto distributed with tail index α (the smallest possible pond is 1). most ponds are small, but the smaller α is, the more often a monster shows up.
the fish draws itself at L × (true median pond ÷ its guess of a typical pond). the dashed outline is its real size. so the fish is always fine. how big it feels comes down to the guess.
the mean brain guesses with the running average of every pond it's seen. for α ≤ 1 the average is infinite, so it never settles: every monster drags it up and the fish shrinks again. for 1 < α ≤ 2 the average exists but the variance doesn't, so it gets there slowly and lurches on the way. above 2 it settles, but the fish still ends up too small, because in a skewed distribution the mean sits above the median. comparing yourself to the average is the bug.
the bayes brain learns α instead. it starts from a weak Gamma(1, 1) prior, and after n ponds it believes Gamma(1 + n, 1 + Σ ln xi). its typical pond is the median of the posterior predictive, exp(b (21/a − 1)), which I checked against two million Monte Carlo draws. one monster barely moves it, it converges to the true median, and the fish ends up its real size.
the plot behind the fish is the log–log CCDF of every pond so far: the share of ponds bigger than each size. a Pareto tail is a straight line there, with slope −α. the pink arrow is the gap between the average pond and the typical one.
ponds are drawn on a log scale, and when a really big one shows up the camera pulls way back so you can see it.
drag α to change the world, flip between the two brains, speed it up, or reseed for new luck. same seed, same ponds, and moving α replays the same luck in a different world.
The fish was fine the whole time. Its estimator was the problem.