This reads to me like a fully generalized argument against numerical modeling.
After all, exponentials are also fake. You can’t be guaranteed to fit a exponential to a data set that has more than two data points. Maybe just fit a linear regression and intuit the rest. Only, the linear regression doesn’t fit perfectly either, so maybe just go on intuition all together.
We know that growth always ends. If you think it won’t look like a sigmoid, perhaps another function that levels out. But the exponential is guaranteed to be wrong in the long run, so why not simply give a wide range for some of the parameters or give multiple models of what could happen?
It’s only an argument against fitting curves to noise. Rather than explain, it turns out there’s already a post that puts this better than I could hope to. I endorse it fully.
Yes, let’s not fit to noise, but fitting to an exponential is also fitting to “noise” in the sense that it’s fitting to a growth pattern that long run is wrong.
If not the sigmoid, there are other functions that produce S-curves that can be used like Gompertz function and the Bass diffusion model. These address some of the issues with sigmoids.
But I also find arguments about modeling being hard fall flat for me. Yes, of course it is hard. Exponential models make one kind of extremely obvious error. Sigmoids make another kind of error. It’s not that hard to at least have a piece-wise model that says “yes, our model predicts growth will end around here” even if that’s hard to fit into a single function, and I see little benefit to not doing that (other than it’s hard and people don’t want to or they are incentivized to do the easy thing and show a model that keeps going up).
The point of a model is to be validly predictive of something. Fitting your exponential is validly predictive of local behaviour more often than not. Often, insanely so.[1] You can directly use the numerical model to make precise and relevant predictions.
Your exponential doesn’t tell you when the trend stops, but it’s not trying to, for one because it’s incapable of modelling that. That’s ok, because that’s not its job.
Fitting a sigmoid doesn’t do this. The majority of times, the only additional thing the result of a sigmoid fit tells you is how an arbitrarily chosen dampening model fits to the arbitrary noise in your data. There’s nothing you can do with that, because it’s not predictive of anything of value.
This doesn’t mean you shouldn’t care about limiting behaviour, or dampening factors. It just means this particular tool, fitting a numerical model to numerical data, isn’t the right tool for reasoning about it.
This reads to me like a fully generalized argument against numerical modeling.
After all, exponentials are also fake. You can’t be guaranteed to fit a exponential to a data set that has more than two data points. Maybe just fit a linear regression and intuit the rest. Only, the linear regression doesn’t fit perfectly either, so maybe just go on intuition all together.
We know that growth always ends. If you think it won’t look like a sigmoid, perhaps another function that levels out. But the exponential is guaranteed to be wrong in the long run, so why not simply give a wide range for some of the parameters or give multiple models of what could happen?
It’s only an argument against fitting curves to noise. Rather than explain, it turns out there’s already a post that puts this better than I could hope to. I endorse it fully.
https://www.lesswrong.com/posts/6tErqpd2tDcpiBrX9/why-sigmoids-are-so-hard-to-predict
Yes, let’s not fit to noise, but fitting to an exponential is also fitting to “noise” in the sense that it’s fitting to a growth pattern that long run is wrong.
If not the sigmoid, there are other functions that produce S-curves that can be used like Gompertz function and the Bass diffusion model. These address some of the issues with sigmoids.
But I also find arguments about modeling being hard fall flat for me. Yes, of course it is hard. Exponential models make one kind of extremely obvious error. Sigmoids make another kind of error. It’s not that hard to at least have a piece-wise model that says “yes, our model predicts growth will end around here” even if that’s hard to fit into a single function, and I see little benefit to not doing that (other than it’s hard and people don’t want to or they are incentivized to do the easy thing and show a model that keeps going up).
The point of a model is to be validly predictive of something. Fitting your exponential is validly predictive of local behaviour more often than not. Often, insanely so.[1] You can directly use the numerical model to make precise and relevant predictions.
Your exponential doesn’t tell you when the trend stops, but it’s not trying to, for one because it’s incapable of modelling that. That’s ok, because that’s not its job.
Fitting a sigmoid doesn’t do this. The majority of times, the only additional thing the result of a sigmoid fit tells you is how an arbitrarily chosen dampening model fits to the arbitrary noise in your data. There’s nothing you can do with that, because it’s not predictive of anything of value.
This doesn’t mean you shouldn’t care about limiting behaviour, or dampening factors. It just means this particular tool, fitting a numerical model to numerical data, isn’t the right tool for reasoning about it.
“I answered that the Gods Of Straight Lines are more powerful than the Gods Of The Copybook Headings, so if you try to use common sense on this problem you will fail.” — Is Science Slowing Down?, Slate Star Codex, https://slatestarcodex.com/2018/11/26/is-science-slowing-down-2/