Age limits do exist: you have to be at least 35 to run for President, at least 30 for Senator, and 25 for Representative. This automatically adds a decade or two to your candidates.

In earlier times, I spent an incredible amount of my mental capacity trying to accurately model those around me. I can count on zero hands the number of people that reciprocated. Even just treating me as real as I treated them would fit on one hand. On the other hand, nearly everyone I talk to does not have "me" as even a possibility in their model.

It just takes a very long time in practice, see "Basins of Attraction" by Ellison.

I've been thinking about something similar, and might write a longer post about it. However, the solution to both is to anneal on your beliefs. Rather than looking at the direct probabilities, look at the logits. You can then raise the temperature, let the population kind of randomize their beliefs, and cool it back down.

See "Solving Multiobjective Game in Multiconflict Situation Based on Adaptive Differential Evolution Algorithm with Simulated Annealing" by Li et. al.

To see how much the minimal point ${\omega }^{\ast }$ contributes to the integral we can integrate it in its vicinity

I think you should be looking at the entire stable island, not just integrating from zero to one. I expect you could get a decent approximation with Lie transform perturbation theory, and this looks similar to the idea of macro-states in condensed matter physics, but I'm not knowledgeable in these areas.

−N∑i=1logp(yi|xi,w)

You have a typo, the equation after Free Energy should start with $NLL\left(w\right)=-\sum \log \left(p\right(y|x_i,w_i\left)\right)q\left(y_i|x_i\right).$

Also the third line should be $\int \cdots +\int$, not minus.

Also, usually people use $\theta$ for model parameters (rather than $w$). I don't know the etymology, but game theorists use the same letter (for "types" = models of players).

Also sometimes when I explain what a hyperphone is well enough for the other person to get it, and then we have a complex conversation, they agree that it would be good. But very small N, like 3 to 5.

It's difficult to understand your writing, and I feel like you could improve in general at communication based on this quote. The concept of a hyperphone isn't that complex---the ability to branch in conversations---so the modifiers "well enough", "complex", and "very small N" make me believe it's only complex because you're unclear.

For example, the blog post you linked to is titled "Hyperphone", yet you never define a hyperphone. I can infer from the section on streaming what you imagine, but that's the second-to-last section!

There's the automorphism

$(x_1,x_2,x_3,\dots ,x_n)\mapsto (x_1,x_1\oplus x_2,x_2\oplus x_3,\dots ,x_{n-1}\oplus x_n)$

which turns a switchy distribution into a sticky one, and vice versa. The two have to be symmetric, so your conclusion cannot be correct.

This means the likelihood distribution over data generated by Steady is closer to the distribution generated by Switchy than to the distribution generated by Sticky.

Their KL divergences are exactly the same. Suppose Baylee's observations are $x_1,\dots ,x_n$. Let $P(x_1,\dots ,x_n)$ be the probability if there's a $p$ chance of switching, and similar for $Q$. By the chain rule,

$\begin{array}{cc}{D}_{KL}\left(P\right(x_1,\dots ,x_n\left)||Q\right(x_1,\dots ,x_n\left)\right)& =D_{KL}\left(P\right(x_1\left)||Q\right(x_1\left)\right)+\sum _{i=1}^{n-1}{D}_{KL}\left(P\right(x_{i+1}|x_i\left)||Q\right(x_{i+1}|x_i\left)\right)\\ & =0+(n-1)[p\log \frac{p}{q}+(1-p\left)\log \frac{1-p}{1-q}\right].\end{array}$

In particular, when either $p$ or $q$ is equal to one half, this divergence is symmetric for the other variable.