J Bostock comments on $500 Bounty Problem: Are (Approximately) Deterministic Natural Latents All You Need?

[J Bostock]

OK so some further thoughts on this: suppose we instead just partition the values of $\Lambda$ directly by something like a clustering algorithm, based on $D_{KL}$ in $P\left[X|\Lambda \right]$ space, and take $\Delta \left(\Lambda \right)$ just be the cluster that $\lambda$ is in:

Assuming we can do it with small clusters, we know that $P\left[X|\Lambda \right]\approx P\left[X|\Delta \right]$ is pretty small, so $D_{KL}\left(P\right[X\left]||P\right[X|\Delta \left]\right)$ is also small.

And if we consider $X_2\leftarrow X_1\to \Lambda$, this tells us that learning $X_1$ restricts us to a pretty small region of $P\left[X_2\right]$ space (since $P\left[X_2|X_1\right]\approx P[X_2|X_1,\Lambda ]$) so $\Delta$ should be approximately deterministic in $X_1$. This second part is more difficult to formalize, though.

Edit: The real issue is whether or not we could have lots of $\Lambda$ values which produce the same distribution over $X_2$ but different distributions over $X_1$, and all be pretty likely given $X_1=x_1$ for some $x_1$. I think this just can’t really happen for probable values of $x_1$, because if these values of $\lambda$ produce the same distribution over $X_2$, but different distributions over $X_1$, then that doesn’t satisfy $X_1\leftarrow X_2\to \Lambda$, and secondly because if they produced wildly different distributions over $X_1$, then that means they can’t all have high values of $P[X_1=x_1|\Lambda =\lambda ]$, and so they’re not gonna have high values of $P[\Lambda =\lambda |X_1=x_1]$.