Probability space has 2 metrics

The speculative proposition that humans might only be using one metric rings true and is compellingly presented.

However, I feel a bit clickbaited by the title, which (to me) implies that probability-space has only two metrics (which isn't true, as the later proposition depends on). Maybe consider changing it to "Probability space has multiple metrics", to avoid confusion?

Note that the closer the probability of something to 0 or to 1, the harder it is evaluate accurately. A simple example: starting with a fair coin and observing a sequence of N heads in a row, what is an unbiased estimate of the coin's bias? Log odds of N heads are -N when starting with a point estimate of a fair coin, which matches the Bayesian updates, so it is reasonable to conclude that the probability of heads is 1-2^(-N), but at the level small enough there are so many other factors that can interfere, the calculation ceases being accurate. Maybe the coin has heads on both sides? Maybe your brain makes you see heads when the coin flip outcome is actually tails? Maybe you are only hallucinating the coin flips? So, if you finally get a tail, reducing the estimated probability of heads, you are able to reject multiple other unlikely possibilities, as well, and it makes sense that one would need less evidence when moving from -N to -N+1 for large N than for small N.

I don’t think I’ve read this view before, or if I have, I’ve forgotten it. Thanks for writing this up!

$P(p1,p2)=|p1-p2|=\mid \frac{1}{e^{p_1}+1}-\frac{1}{e^{p_2}+1}|$

I think this should have b instead of p: $P(p1,p2)=|p1-p2|=\mid \frac{1}{e^{b_1}+1}-\frac{1}{e^{b_2}+1}|$

Awesome idea! I think there might be something here, but I think the difference between "no chance" and "0.01% chance" is more of a discrete change from not tracking something to tracking it. We might also expect neglect of "one in a million" vs "one in a trillion" in both updates and decision-making, which causes a mistake opposite that predicted by this model in the case of decision-making.

I'm pretty sure this point has been made here before, but, hey, it's worth repeating, no? :)

I like the theory. How would we test it?

We have a fairly good idea of how people weight decisions based on probabilities via offering different bets and seeing which ones get chosen.

I don't know how much quantification has been done on incorrect Bayesian updates. Could one suggest trades where one is given options one of which has been recommended by an "expert" who has made the correct prediction to a 50:50 question on a related topic x times in a row. How much do people adjust based on the evidence of the expert? This doesn't sound perfect to me, maybe someone else has a better version or maybe people are already doing this research?!