The radioactive decay example specifically? Fit A and k (e.g. by nonlinear least squares) and then use the Jacobian to obtain the approximate covariance matrix. The diagnonal elements of that matrix give you the standard error estimates.
I think we are just coming at this from different angles. I do understand and agree that we are estimating the parameters of the fit curves.
> That already makes strong modeling assumptions (usually including IID, Gaussian noise, etc.,) to get the parameter estimates in the first place
You lose me here - I don't agree with "usually". I guess you're thinking of examples where you are sampling from a population and estimating features of that population. There's nothing wrong with that, but that is a much smaller domain than curve fitting in general.
If you give me a set of x and y, I can fit a parametric curve that tries to minimises the average squared distance between fit and observed values of y without making any assumptions whatsoever. This is a purely mechanical, non-stochastic procedure.
For example, if you give me the points {(0,0), (1,1), (2,4), (3,9)} and the curve y = a x^b, then I'm going to fit a=1, b=2, and I certainly don't need to assume anything about the data generating process to do so. However there is no concept of a confidence interval in this example - the estimates are the estimates, the residual error is 0, and that is pretty much all that can be said.
If you go further and tell me that each of these pairs (x,y) is randomly sampled, or maybe the x is fixed and the y is sampled, then I can do more. But that is often not the case.
I don’t think you can do anything sensible here without making much stronger modelling assumptions. A vanilla non-parametric bootstrap is only valid under a very specific generative story: IID sampling from a population. Many (most?) curve-fitting problems won't satisfy that.
For example, suppose you measure the decay of a radioactive source at fixed times t = 0,1,2,... and fit y = A e^{-kt}. The only randomness is small measurement error with, say, SD = 0.5. The bootstrap sees the huge spread in the y-values that comes from the deterministic decay curve itself, not from noise. It interprets that structural variation as sampling variability and you end up with absurdly wide bootstrap confidence intervals that have nothing to do with the actual uncertainty in the experiment.
As long as UK taxes are flow-based and not stock-based, it seems a bit silly to base analysis on a stock-based denominator like the number of millionaires.
I wouldn’t call the embedding layer "separate" from the LLM. It’s learned jointly with the rest of the network, and its dimensionality is one of the most fundamental architectural choices. You’re right though that, in principle, you can pick an embedding size independent of other hyperparameters like number of layers or heads, so I see where you're coming from.
However the embedding dimension sets the rank of the token representation space. Each layer can transform or refine those vectors, but it can’t expand their intrinsic capacity. A tall but narrow network is bottlenecked by that width. Width-first scaling tends to outperform pure depth scaling, you want enough representational richness per token before you start stacking more layers of processing.
So yeah, embedding size doesn’t have to scale up in lockstep with model size, but in practice it usually does, because once models grow deeper and more capable, narrow embeddings quickly become the limiting factor.