Why's that? They seem to be going for AGI, can afford to invest billions if Zuckerberg chooses, their effort is led by one of the top AI researchers and they have produced some systems that seem impressive (at least to me). If you wanted to cover your bases, wouldn't it make sense to include them? Though 3-5% may be a bit much (but I also think it's a bit much for the listed companies besides MS and Google). Or can a strong argument be made for why, if AGI were attained in the near term, they wouldn't be the ones to profit from it?

• Invest like 3-5% of my portfolio into each of Nvidia, TSMC, Microsoft, Google, ASML and Amazon

Should Meta be in the list? Are the big Chinese tech companies considered out of the race?

Do you mean you'd be adding the probability distribution with that covariance matrix on top of the mean prediction from f, to make it a probabilistic prediction? I was talking about deterministic predictions before, though my text doesn't make that clear. For probabilistic models, yes adding an uncertainty distribution may make result in non-zero likelihoods. But if we know the true dynamics are deterministic (pretend there's no quantum effects, which are largely irrelevant for our prediction errors for systems in the classical physics domain), then we still know the model is not true, and so it seems difficult to interpret p if we were to do Bayesian updating.

Likelihoods are also not obviously (to me) very good measures of model quality for chaotic systems, either - in these cases we know that even if we had the true model, its predictions would diverge from reality due to errors in the initial condition estimates, but it would trace out the correct attractor - and its the attractor geometry (conditional on boundary conditions) that we'd seem to really want to assess. Perhaps then it would have a higher likelihood than every other model, but it's not obvious to me, and it's not obvious that there's not a better metric for leading to good inferences when we don't have the true model.

Basically the logic that says to use Bayes for deducing the truth does not seem to carry over in an obvious way (to me) to the case when we want to predict but can't use the true model.

Yes I'd selected that because I thought it might get it to work. And now I've unselected it, it seems to be working. It's possible this was a glitch somewhere or me just being dumb before I guess.

I wonder whether the models are so coarse that the cyclones that do emerge are in a sense the minimum size.

It's not my area, but I don't think that's the case. My impression is that part of what drives very high wind speeds in the strongest hurricanes is convection on the scale of a few km in the eyewall, so models with that sort of spatial resolution can generate realistically strong systems, but that's ~20x finer than typical climate model resolutions at the moment, so it will be a while before we can simulate those systems routinely (though, some argue we could do it if we had a computer costing a few billion dollars).

do you know what discretization methods are typically used for the fluid dynamics?

There's a mixture - finite differencing used to be used a lot but seems to be less common now, semi-Lagrangian advection seems to have taken over from that in models that used it, then some work by doing most of the computations in spectral space and neglecting the smallest spatial scales. Recently newer methods have been developed to work better on massively parallel computers. It's not my area, though, so I can't give a very expert answer - but I'm pretty sure the people working on it think hard about trying to not smooth out intense structures (though, that has to be balanced against maintaining numerical stability).

I'm using Chrome 80.0.3987.163 in Mac OSX 10.14.6. But I also tried it in Firefox and didn't get formatting options. But maybe I'm just doing the wrong thing...

Thanks, yes this is very relevant to thinking about climate modelling, with the dominant paradigm being that we can separately model phenomena above and below the resolved scale - there's an ongoing debate, though, about whether a different approach would work better, and it gets tricky when the resolved scale gets close to the size of important types of weather system.

climate models are already "low-level physics" except that "low-level" means coarse aggregates of climate/weather measurements that are so big that they don't include tropical cyclones!

Just as as aside, a typical modern climate model will simulate tropical cyclones as emergent phenomena from the coarse-scale fluid dynamics, albeit not enough of the most intense ones. Though, much smaller tropical thunderstorm-like systems are much more crudely represented.

Thanks again.

I think I need to think more about the likelihood issue. I still feel like we might be thinking about different things - when you say "a deterministic model which uses fundamental physics", this would not be in the set of models that we could afford to run to make predictions for complex systems. For the models we could afford to run, it seems to me that no choice of initial conditions would lead them to match the data we observe, except by extreme coincidence (analogous to a simple polynomial just happening to pass through all the datapoints produced by a much more complex function).

I've gone through Jaynes' paper now from the link you gave. His point about deciding what macroscopic variables matter is well-made. But you still need a model of how the macroscopic variables you observe relate to the ones you want to predict. In modelling atmospheric processes, simple spatial averaging of the fluid dynamics equations over resolved spatial scales gets you some way, but then changing the form of the function relating the future to present states ("adding representations of processes" as I put it before) adds additional skill. And Jaynes' paper doesn't seem to say how you should choose this function.