Overall, I really want someone who is a proponent of the hyperbolic model to explain to me why this model is so popular, because to me it seems obviously wrong. I’d be happy to schedule a call with someone just for this purpose.
Could you describe an alternative model that you think has a less bad fit to the historical data?
(If you are just saying this on the basis of qualitative data, I disagree but don’t think it’s going to be helpful to try to resolve here.)
I think mixture of exponentials is a significantly worse fit and have tried to do this exercise myself. If you disagree then I’d be quite happy for you to describe a probability distribution over time series which you think is better than the natural stochastic hyperbolic model (and I will bring the stochastic hyperbolic model). By default I won’t include an explicit model of autocorrelations unless you do, and if I did it would be a very simple representation with a reasonable prior over any hyperparameters involved rather than hard-coding them.
You can pick the times and places where you think the data is good enough that it’s worth modeling, and even provide your own time series if you think the hyperbolic growth proponents are making a mistake about which data to trust. E.g. I’d be fine if you want to do it in just the UK or europe, or using best guess global time series, or using best guess global time series since 1000 AD if you don’t trust them before that (since they are quite wild guesses). Or if you want to fit to a variety of different local time series. Whichever.
I think if this exercise favored a mixture of exponentials that would be a significant update for me. So far every time I’ve looked into it a mixture of exponentials has seemed significantly worse, which is a large part of how I ended up with this view (e.g. it basically ends up with totally different hyperparameters for each population studied, fitting it up through year X gives you bad predictions about year X+100, when I do the calculation formally it looks bad...). But I think those exercises have been hamstrung by not having any proponent of the mixtures of exponential view write down a formal model that could be evaluated without hindsight bias.
Could you describe an alternative model that you think has a less bad fit to the historical data?
(If you are just saying this on the basis of qualitative data, I disagree but don’t think it’s going to be helpful to try to resolve here.)
I think mixture of exponentials is a significantly worse fit and have tried to do this exercise myself. If you disagree then I’d be quite happy for you to describe a probability distribution over time series which you think is better than the natural stochastic hyperbolic model (and I will bring the stochastic hyperbolic model). By default I won’t include an explicit model of autocorrelations unless you do, and if I did it would be a very simple representation with a reasonable prior over any hyperparameters involved rather than hard-coding them.
You can pick the times and places where you think the data is good enough that it’s worth modeling, and even provide your own time series if you think the hyperbolic growth proponents are making a mistake about which data to trust. E.g. I’d be fine if you want to do it in just the UK or europe, or using best guess global time series, or using best guess global time series since 1000 AD if you don’t trust them before that (since they are quite wild guesses). Or if you want to fit to a variety of different local time series. Whichever.
I think if this exercise favored a mixture of exponentials that would be a significant update for me. So far every time I’ve looked into it a mixture of exponentials has seemed significantly worse, which is a large part of how I ended up with this view (e.g. it basically ends up with totally different hyperparameters for each population studied, fitting it up through year X gives you bad predictions about year X+100, when I do the calculation formally it looks bad...). But I think those exercises have been hamstrung by not having any proponent of the mixtures of exponential view write down a formal model that could be evaluated without hindsight bias.