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.
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.