p = (n^c * (c + 1)) / (2^c * n)

As far as I know, this is unpublished in the literature. It's a pretty obscure use case, so that's not surprising. I have doubts I'll ever get around to publishing the paper I wanted to write that uses this in an activation function to replace softmax in neural nets, so it probably doesn't matter much if I show it here.

So, my main idea is that the principle of maximum entropy aka the principle of indifference suggests a prior of 1/n where n is the number of possibilities or classes. P x 2 - 1 leads to p = 0.5 for c = 0. What I want is for c = 0 to lead to p = 1/n rather than 0.5, so that it works in the multiclass cases where n is greater than 2.

Correlation space is between -1 and 1, with 1 being the same (definitely true), -1 being the opposite (definitely false), and 0 being orthogonal (very uncertain). I had the idea that you could assume maximum uncertainty to be 0 in correlation space, and 1/n (the uniform distribution) in probability space.

I tried asking ChatGPT, Gemini, and Claude to come up with a formula that converts between correlation space to probability space while preserving the relationship 0 = 1/n. I came up with such a formula a while back, so I figure it shouldn't be hard. They all offered formulas, all of which were shown to be very much wrong when I actually graphed them to check.

I was not aware of these. Thanks!

Thanks for the clarifications. My naive estimate is obviously just a simplistic ballpark figure using some rough approximations, so I appreciate adding some precision.

Also, even if we can train and run a model the size of the human brain, it would still be many orders of magnitude less energy efficient than an actual brain. Human brains use barely 20 watts. This hypothetical GPU brain would require enormous data centres of power, and each H100 GPU uses 700 watts alone.

I've been looking at the numbers with regards to how many GPUs it would take to train a model with as many parameters as the human brain has synapses. The human brain has 100 trillion synapses, and they are sparse and very efficiently connected. A regular AI model fully connects every neuron in a given layer to every neuron in the previous layer, so that would be less efficient.

The average H100 has 80 GB of VRAM, so assuming that each parameter is 32 bits, then you have about 20 billion per GPU. So, you'd need 10,000 GPUs to fit a single instance of a human brain in RAM, maybe. If you assume inefficiencies and need to have data in memory as well you could ballpark another order of magnitude so 100,000 might be needed.

For comparison, it's widely believed that OpenAI trained GPT4 on about 10,000 A100s that Microsoft let them use from their Azure supercomputer, most likely the one listed as third most powerful in the world by the Top500 list.

Recently though, Microsoft and Meta have both moved to acquire more GPUs that put them in the 100,000 range, and Elon Musk's X.ai recently managed to get a 100,000 H100 GPU supercomputer online in Memphis.

So, in theory at least, we are nearly at the point where they can train a human brain sized model in terms of memory. However, keep in mind that training such a model would take a ton of compute time. I haven't done to calculations yet for FLOPS so I don't know if it's feasible yet.

Just some quick back of the envelope analysis.

I ran out of the usage limit for GPT-4o (seems to just be 10 prompts every 5 hours) and it switched to GPT-4o-mini. I tried asking it the Alpha Omega question and it made some math nonsense up, so it seems like the model matters for this for some reason.

So, a while back I came up with an obscure idea I called the Alpha Omega Theorem and posted it on the Less Wrong forums. Given how there's only one post about it, it shouldn't be something that LLMs would know about. So in the past, I'd ask them "What is the Alpha Omega Theorem?", and they'd always make up some nonsense about a mathematical theory that doesn't actually exist. More recently, Google Gemini and Microsoft Bing Chat would use search to find my post and use that as the basis for their explanation. However, I only have the free version of ChatGPT and Claude, so they don't have access to the Internet and would make stuff up.

A couple days ago I tried the question on ChatGPT again, and GPT-4o managed to correctly say that there isn't a widely known concept of that name in math or science, and basically said it didn't know. Claude still makes up a nonsensical math theory. I also today tried telling Google Gemini not to use search, and it also said it did not know rather than making stuff up.

I'm actually pretty surprised by this. Looks like OpenAI and Google figured out how to reduce hallucinations somehow.