I think it’s worth fitting sum of exponentials (with arbitrarily tuned CES parameters) to continuous random processes to get intuition for what the result looks like. If you draw 10 samples (e.g. from the model used by David Roodman here) I think they will look about as sum-of-exponentials-y as our actual history.
Roodman himself admits that his model is surprised by the data in the industrial phase. It first underestimates growth after ~ 1820 and then consistently overestimates it after ~ 1990 (most likely because it’s expecting something resembling a singularity that doesn’t come). That’s exactly how I would expect such a model to fail if my model is correct.
That said, I think the experiment is worth doing and I’ll try to run some Monte Carlo simulations to examine this more closely when I have the time.
If that’s right then it comes down to the qualitative data, where I disagree (I don’t think it’s at all clear that something discontinuous is happening) but it seems harder to settle.
It’s harder to settle in the comments for sure. All I can say is that I’m pretty convinced of it—I might write a complementary piece to this one later to try to explain why I think that way.
(I don’t think the “industrial revolution is a new exponential spreading” fits the data very well—the industrial revolution phase is much faster than growth in the places where you claim the IR started, it’s like 12%/year vs <1% per year for Britain at the start of the IR!)
Growth rates in the industrial phase are usually around 3%/year. The reason many countries had much faster growth than this is because they are catching up from behind: they were late to the transition, which makes explosive growth easier to achieve until you’ve caught up.
If you don’t mean catch-up growth by countries like China and Japan, then I’m mystified by your 12%/year figure. Where does this estimate come from?
But even if you think the industrial mode is 3%, it seems like “started in Britain and spread out” doesn’t explain the gradual nature of the transition. I don’t have numbers right now but I think the UK itself didn’t reach that growth rate until the 20th century. If there was a phase change it was presumably somewhere closer to 1700?
Roodman himself admits that his model is surprised by the data in the industrial phase.
I agree that the errors in this model are autocorrelated (though I think the mixture of exponentials model avoids this, to the extent it does, mostly because it has many more free parameters---8 is a huge number of parameters, so it’s super unsurprising they can absorb most of the autocorrelation).
I think the “noise” reflects stuff like “how well are institutions doing?” and “does this part of the technology landscape happen to have somewhat faster or slower progress?” those factors aren’t totally independent from one time to another. But I think the hypothesis “actually we’re looking at a few phase transitions” is not really supported over “it’s pretty noisy with mild autocorrelation.”
(My further guess is that if you actually do a bayesian fit, the model class in David Roodman’s data probably fits the data better despite completely giving up on the autocorrelations. E.g. if every 100 years you fit both models to history then use both of them to make a prediction about GDP growth over the next 100 years, you will get more total log loss by using the CES mixture of exponentials.)
I don’t have numbers right now but I think the UK itself didn’t reach that growth rate until the 20th century. If there was a phase change it was presumably somewhere closer to 1700?
It gives 0.6% average growth from 1476 to 1781 and then 2% from 1781 to early 1900s. Though the rate isn’t very constant within either period, closer to 1% at 1781 itself and then hitting 3%+ later in the 20th century.
I’m not sure what the phase transition view is saying about these cases---0.6%/year is incredibly fast growth (that’s a ~100 year doubling time, contrasted with a >1000 year doubling time for other agricultural societies); has the phase transition happened or not by 1476? (You could attribute some to catch-up growth after the plague, but even peak to peak and completely ignoring the plague you have fast growth well before 1781.)
I’m not sure what the phase transition view is saying about these cases---0.6%/year is incredibly fast growth (that’s a ~100 year doubling time, contrasted with a >1000 year doubling time for other agricultural societies); has the phase transition happened or not by 1476? (You could attribute some to catch-up growth after the plague, but even peak to peak and completely ignoring the plague you have fast growth well before 1781.)
My guess would be sometime around the English Civil War. The phase transition definitely hadn’t happened by 1476 - I would put the date sometime in the first half of the 17th century for when it starts, and I think it ends roughly when the Napoleonic Wars are over.
That said, this data is less clear than I would have liked. The GDP per capita plot for the UK shows a very clear break with trend in the middle of the 17th century and this was one of the data points that had convinced me in the past that something discontinuous was going on, but actually this could be the Civil War messing up what otherwise would have been a smoother trend given that this uptick is not visible in the GDP plots.
Combined with the knowledge that the English Civil War happened around the time of the GDP per capita trend deviation, I agree that these plots are evidence against the phase transition model and in favor of a smoother (such as a hyperbolic) model.
Given all the evidence you’ve presented so far, I now think the picture is less clear than I’d previously thought it was.
But even if you think the industrial mode is 3%, it seems like “started in Britain and spread out” doesn’t explain the gradual nature of the transition. I don’t have numbers right now but I think the UK itself didn’t reach that growth rate until the 20th century. If there was a phase change it was presumably somewhere closer to 1700?
Yes, this by itself doesn’t explain it, though it plays a big role in the account for GWP. This is not inconsistent with a phase transition story, but it can be inconsistent with particular versions of it like a direct sum of exponentials model. I think the industrial phase transition was indeed more gradual even disregarding concerns of heterogeneity.
I’m deliberately being rather vague in what the content of the phase transitions are because I don’t think you need a precise understanding of them for the predictions I make in the post. For other predictions, though, an exact specification of a model could be quite valuable.
I agree that the errors in this model are autocorrelated (though I think the mixture of exponentials model avoids this, to the extent it does, mostly because it has many more free parameters---8 is a huge number of parameters, so it’s super unsurprising they can absorb most of the autocorrelation).
You can probably specify versions of the exponential model with much fewer free parameters. One sketch:
An exponential distribution for the mean number of doublings in a new phase (one free parameter)
A lognormal distribution for the growth speed factor of a new phase over the previous phase (two free parameters)
A lognormal distribution for the gross growth factor in any given year (log-mean fixed by the phase, a new free parameter for the ratio of mean to standard deviation)
One parameter to control in some fashion how smooth the transition from one phase to another is
That’s five free parameters for a model of the “phase transition” idea that I think is quite reasonable.
(My further guess is that if you actually do a bayesian fit, the model class in David Roodman’s data probably fits the data better despite completely giving up on the autocorrelations. E.g. if every 100 years you fit both models to history then use both of them to make a prediction about GDP growth over the next 100 years, you will get more total log loss by using the CES mixture of exponentials.)
I’m uncertain about whether this is true or not. I think it could be true and it would be interesting to try it, but even if it were true it would only make me think that this is because the hyperbolic model is doing well on metrics that help it earn log score and not so well on metrics that I care about in this post.
I’d be interested to see what happens if you score one exponential based model and one stochastic hyperbolic model on answering the following question:
Let T be the time it took until now for GWP to double. What’s the probability that GWP will increase by more than a factor of 8 in the next T years?
My guess is that at parameter parity a model based on phase transitions will do better on answering this type of question compared to a hyperbolic model even though it might actually have worse log score.
If the log score of the hyperbolic model was higher than the phase change model, hopefully you would at least see why some people would lean towards accepting it. I think the kind of data you are offering against the hyperbolic model (like “how does it explain X?”) can be pretty well explained as “it’s a low-resolution stochastic model, there’s a lot of noise.” If in fact the model is getting a better log probability than alternative theories, it’s not clear on what grounds you’d reject it.
I think it’s very reasonable to define a model over parameters per phase. To the extent that it makes the model much more complicated than the hyperbolic growth model, I’d be increasingly inclined to just add the simple autocorrelation term with a prior over strength of autocorrelation.
I’d be interested to see what happens if you score one exponential based model and one stochastic hyperbolic model on answering the following question:
It seems like the answer is almost always no and the question is just how low a probability you give. We have at most 1 yes datapoint for that question (which is based on almost complete guesswork about early populations, e.g. see Ben’s critique of data behind continuous acceleration), and it’s going to be extremely sensitive to hyperparameter choices. So overall I’m a bit skeptical of getting something out of that comparison.
You could certainly have a take like “looking at the historical performance of models is not a helpful way to discriminate amongst them, given the small amount of highly unreliable data,” but hopefully that should just make you more rather than less sympathetic to someone who adopts a different model.
First, thanks for participating in this thread. I think I now have a better sense of where the proponents of the hyperbolic model are coming from thanks to your comments.
If the log score of the hyperbolic model was higher than the phase change model, hopefully you would at least see why some people would lean towards accepting it. I think the kind of data you are offering against the hyperbolic model (like “how does it explain X?”) can be pretty well explained as “it’s a low-resolution stochastic model, there’s a lot of noise.” If in fact the model is getting a better log probability than alternative theories, it’s not clear on what grounds you’d reject it.
I think it’s very reasonable to define a model over parameters per phase. To the extent that it makes the model much more complicated than the hyperbolic growth model, I’d be increasingly inclined to just add the simple autocorrelation term with a prior over strength of autocorrelation.
In this case I think we wouldn’t disagree: I also think the sum of exponentials model doesn’t capture the details of the phase transitions I’m proposing well at all. I think actually designing a good model for the phase transition idea is a challenge and I haven’t thought as much about it as I perhaps should have; I’ll get back to this thread at some point in the future when I think I have a good enough model for that.
I think this kind of stochastic model does have good ingredients—in particular, even if the phase transition idea is totally correct, I think a hyperbolic model probably gets a better log score on how its dynamics of internal spread end up working out. I just don’t think the model is good for answering the particular kind of question I want to answer in this post.
It seems like the answer is almost always no and the question is just how low a probability you give. We have at most 1 yes datapoint for that question (which is based on almost complete guesswork about early populations, e.g. see Ben’s critique of data behind continuous acceleration), and it’s going to be extremely sensitive to hyperparameter choices. So overall I’m a bit skeptical of getting something out of that comparison.
I think we have two yes datapoints: they are the time periods around the two phase transitions I’m proposing. I agree there’s a lot of guesswork here but I think the range of uncertainty is not so large that we don’t actually know the answer to my question in these cases.
You could certainly have a take like “looking at the historical performance of models is not a helpful way to discriminate amongst them, given the small amount of highly unreliable data,” but hopefully that should just make you more rather than less sympathetic to someone who adopts a different model.
Sure. After listening to your comments and going through some of the data myself, I’ve changed my mind about this. I still think that I wouldn’t trust the hyperbolic model with the specific kind of prediction I mentioned, however, and this just happens to be the relevant kind of prediction in this context. The issue with tuning the autocorrelation term to match recent history seems particularly dangerous to me if we’re forecasting something about the next 50 or 100 years.
Roodman himself admits that his model is surprised by the data in the industrial phase. It first underestimates growth after ~ 1820 and then consistently overestimates it after ~ 1990 (most likely because it’s expecting something resembling a singularity that doesn’t come). That’s exactly how I would expect such a model to fail if my model is correct.
That said, I think the experiment is worth doing and I’ll try to run some Monte Carlo simulations to examine this more closely when I have the time.
It’s harder to settle in the comments for sure. All I can say is that I’m pretty convinced of it—I might write a complementary piece to this one later to try to explain why I think that way.
Growth rates in the industrial phase are usually around 3%/year. The reason many countries had much faster growth than this is because they are catching up from behind: they were late to the transition, which makes explosive growth easier to achieve until you’ve caught up.
If you don’t mean catch-up growth by countries like China and Japan, then I’m mystified by your 12%/year figure. Where does this estimate come from?
12% is the estimate from Robin’s model.
But even if you think the industrial mode is 3%, it seems like “started in Britain and spread out” doesn’t explain the gradual nature of the transition. I don’t have numbers right now but I think the UK itself didn’t reach that growth rate until the 20th century. If there was a phase change it was presumably somewhere closer to 1700?
I agree that the errors in this model are autocorrelated (though I think the mixture of exponentials model avoids this, to the extent it does, mostly because it has many more free parameters---8 is a huge number of parameters, so it’s super unsurprising they can absorb most of the autocorrelation).
I think the “noise” reflects stuff like “how well are institutions doing?” and “does this part of the technology landscape happen to have somewhat faster or slower progress?” those factors aren’t totally independent from one time to another. But I think the hypothesis “actually we’re looking at a few phase transitions” is not really supported over “it’s pretty noisy with mild autocorrelation.”
(My further guess is that if you actually do a bayesian fit, the model class in David Roodman’s data probably fits the data better despite completely giving up on the autocorrelations. E.g. if every 100 years you fit both models to history then use both of them to make a prediction about GDP growth over the next 100 years, you will get more total log loss by using the CES mixture of exponentials.)
Here’s a citation on britain GDP growth (but haven’t checked them for reasonableness), see figures 2 and 7: https://academic.oup.com/ereh/article/21/2/141/3044162.
It gives 0.6% average growth from 1476 to 1781 and then 2% from 1781 to early 1900s. Though the rate isn’t very constant within either period, closer to 1% at 1781 itself and then hitting 3%+ later in the 20th century.
I’m not sure what the phase transition view is saying about these cases---0.6%/year is incredibly fast growth (that’s a ~100 year doubling time, contrasted with a >1000 year doubling time for other agricultural societies); has the phase transition happened or not by 1476? (You could attribute some to catch-up growth after the plague, but even peak to peak and completely ignoring the plague you have fast growth well before 1781.)
My guess would be sometime around the English Civil War. The phase transition definitely hadn’t happened by 1476 - I would put the date sometime in the first half of the 17th century for when it starts, and I think it ends roughly when the Napoleonic Wars are over.
That said, this data is less clear than I would have liked. The GDP per capita plot for the UK shows a very clear break with trend in the middle of the 17th century and this was one of the data points that had convinced me in the past that something discontinuous was going on, but actually this could be the Civil War messing up what otherwise would have been a smoother trend given that this uptick is not visible in the GDP plots.
Combined with the knowledge that the English Civil War happened around the time of the GDP per capita trend deviation, I agree that these plots are evidence against the phase transition model and in favor of a smoother (such as a hyperbolic) model.
Given all the evidence you’ve presented so far, I now think the picture is less clear than I’d previously thought it was.
Yes, this by itself doesn’t explain it, though it plays a big role in the account for GWP. This is not inconsistent with a phase transition story, but it can be inconsistent with particular versions of it like a direct sum of exponentials model. I think the industrial phase transition was indeed more gradual even disregarding concerns of heterogeneity.
I’m deliberately being rather vague in what the content of the phase transitions are because I don’t think you need a precise understanding of them for the predictions I make in the post. For other predictions, though, an exact specification of a model could be quite valuable.
You can probably specify versions of the exponential model with much fewer free parameters. One sketch:
An exponential distribution for the mean number of doublings in a new phase (one free parameter)
A lognormal distribution for the growth speed factor of a new phase over the previous phase (two free parameters)
A lognormal distribution for the gross growth factor in any given year (log-mean fixed by the phase, a new free parameter for the ratio of mean to standard deviation)
One parameter to control in some fashion how smooth the transition from one phase to another is
That’s five free parameters for a model of the “phase transition” idea that I think is quite reasonable.
I’m uncertain about whether this is true or not. I think it could be true and it would be interesting to try it, but even if it were true it would only make me think that this is because the hyperbolic model is doing well on metrics that help it earn log score and not so well on metrics that I care about in this post.
I’d be interested to see what happens if you score one exponential based model and one stochastic hyperbolic model on answering the following question:
My guess is that at parameter parity a model based on phase transitions will do better on answering this type of question compared to a hyperbolic model even though it might actually have worse log score.
If the log score of the hyperbolic model was higher than the phase change model, hopefully you would at least see why some people would lean towards accepting it. I think the kind of data you are offering against the hyperbolic model (like “how does it explain X?”) can be pretty well explained as “it’s a low-resolution stochastic model, there’s a lot of noise.” If in fact the model is getting a better log probability than alternative theories, it’s not clear on what grounds you’d reject it.
I think it’s very reasonable to define a model over parameters per phase. To the extent that it makes the model much more complicated than the hyperbolic growth model, I’d be increasingly inclined to just add the simple autocorrelation term with a prior over strength of autocorrelation.
It seems like the answer is almost always no and the question is just how low a probability you give. We have at most 1 yes datapoint for that question (which is based on almost complete guesswork about early populations, e.g. see Ben’s critique of data behind continuous acceleration), and it’s going to be extremely sensitive to hyperparameter choices. So overall I’m a bit skeptical of getting something out of that comparison.
You could certainly have a take like “looking at the historical performance of models is not a helpful way to discriminate amongst them, given the small amount of highly unreliable data,” but hopefully that should just make you more rather than less sympathetic to someone who adopts a different model.
First, thanks for participating in this thread. I think I now have a better sense of where the proponents of the hyperbolic model are coming from thanks to your comments.
In this case I think we wouldn’t disagree: I also think the sum of exponentials model doesn’t capture the details of the phase transitions I’m proposing well at all. I think actually designing a good model for the phase transition idea is a challenge and I haven’t thought as much about it as I perhaps should have; I’ll get back to this thread at some point in the future when I think I have a good enough model for that.
I think this kind of stochastic model does have good ingredients—in particular, even if the phase transition idea is totally correct, I think a hyperbolic model probably gets a better log score on how its dynamics of internal spread end up working out. I just don’t think the model is good for answering the particular kind of question I want to answer in this post.
I think we have two yes datapoints: they are the time periods around the two phase transitions I’m proposing. I agree there’s a lot of guesswork here but I think the range of uncertainty is not so large that we don’t actually know the answer to my question in these cases.
Sure. After listening to your comments and going through some of the data myself, I’ve changed my mind about this. I still think that I wouldn’t trust the hyperbolic model with the specific kind of prediction I mentioned, however, and this just happens to be the relevant kind of prediction in this context. The issue with tuning the autocorrelation term to match recent history seems particularly dangerous to me if we’re forecasting something about the next 50 or 100 years.