I think you can replace the first instance of “are statistically independent” with “are statistically independent and identically distributed” & improve clarity.
IMO, your argument needs work if you want it to be more than an intuition pump. If the question is the existence or nonexistence of particular clusters, you are essentially assuming what you need to prove in this post. Plus, the existence or nonexistence of clusters is a “choice of ontology” question which doesn’t necessarily have a single correct answer.
You’re also fuzzing things by talking about discrete distributions here, then linking to Eliezer’s discussion of continuous latent variables (“intelligence”) without noting the difference. And: If a number of characteristics have been observed to co-vary, this isn’t sufficient evidence for any particular causal mechanism. Correlation isn’t causation. As I pointed out in this essay, it’s possible there’s some latent factor like the ease of obtaining calories in an organism’s environment which explains interspecies intelligence differences but doesn’t say anything about the “intelligence” of software.
replace the first instance of “are statistically independent” with “are statistically independent and identically distributed”
Done, thanks!
talking about discrete distributions here, then linking to Eliezer’s discussion of continuous latent variables (“intelligence”) without noting the difference
The difference doesn’t seem relevant to the narrow point I’m trying to make? I was originally going to use multivariate normal distributions with different means, but then decided to just make up “peaked” discrete distributions in order to keep the arithmetic simple.
I agree with your other two points (mostly—I don’t feel that the distinction between discrete and continuous variables is important for Zack’s argument so it seems fine to gloss over it) but I disagree with the first.
In order to be able to simply multiply likelihood ratios, the sufficient fact is that they’re statistically independent. In this toy model, they also happen to be identically distributed, but I think it’s clear from context that Zack would like to apply his argument to a variety of situations where the different dimensions have different distributions. You’re suggesting replacing “X, therefore Z” with “X and Y, therefore Z”, when in fact X->Z, and it is not the case that Y->Z.
Good post. Some feedback:
I think you can replace the first instance of “are statistically independent” with “are statistically independent and identically distributed” & improve clarity.
IMO, your argument needs work if you want it to be more than an intuition pump. If the question is the existence or nonexistence of particular clusters, you are essentially assuming what you need to prove in this post. Plus, the existence or nonexistence of clusters is a “choice of ontology” question which doesn’t necessarily have a single correct answer.
You’re also fuzzing things by talking about discrete distributions here, then linking to Eliezer’s discussion of continuous latent variables (“intelligence”) without noting the difference. And: If a number of characteristics have been observed to co-vary, this isn’t sufficient evidence for any particular causal mechanism. Correlation isn’t causation. As I pointed out in this essay, it’s possible there’s some latent factor like the ease of obtaining calories in an organism’s environment which explains interspecies intelligence differences but doesn’t say anything about the “intelligence” of software.
Done, thanks!
The difference doesn’t seem relevant to the narrow point I’m trying to make? I was originally going to use multivariate normal distributions with different means, but then decided to just make up “peaked” discrete distributions in order to keep the arithmetic simple.
I agree with your other two points (mostly—I don’t feel that the distinction between discrete and continuous variables is important for Zack’s argument so it seems fine to gloss over it) but I disagree with the first.
In order to be able to simply multiply likelihood ratios, the sufficient fact is that they’re statistically independent. In this toy model, they also happen to be identically distributed, but I think it’s clear from context that Zack would like to apply his argument to a variety of situations where the different dimensions have different distributions. You’re suggesting replacing “X, therefore Z” with “X and Y, therefore Z”, when in fact X->Z, and it is not the case that Y->Z.