> Like I said, there is no way to test for how well it will extrapolate...
> Of course there is. It is standard practice in machine learning, see cross-validation.
I agree with Graham: you _can_ get some information about how well a method of extrapolating the QBO works, using only the data we have. And if one wants to claim one knows how to extrapolate the QBO back to 1880, one needs whatever evidence one can get! - even though we'll never know for sure.
On the other hand, one can merely claim one has a method of retrodicting the SOI back to 1880. In this approach - which only differs in the _claims one makes_, extrapolating the QBO back to 1880 is treated as just part of a bigger machine. The machine is justified solely by its ability to retrodict the SOI.
The difference is this:
In the latter approach, all the parameters used to extrapolate the QBO back to 1880 count as adjustable parameters in the SOI retrodiction. So, if we use something like the [Akiake information criterion](https://en.wikipedia.org/wiki/Akaike_information_criterion) to see how good the SOI retrodiction is, these parameters "cost us".
I hope I'm making myself clear here.