And not being sure how to think about Chess ratings and whether this discontinuity is better thought of as linear or exponential or what.
Are you talking about the overall trend, or the discontinuity itself?
It doesn’t seem like it would make sense to talk about the discontinuity itself as linear / exponential, since it’s defined by just two points: the point before the jump, and the point after the jump. You could fit a line through two points, you could fit an exponential through two points, you could fit anything!
(If you had a trend that switched from being linear to being exponential, that would be a different story. But this graph doesn’t look like that to me.)
Are you talking about the overall trend, or the discontinuity itself?
I was mostly talking about the overall trend, although I have additional thoughts on your point about the point-of-discontinuity.
(epistemic status: a bit outside my comfort zone. I feel confident but wouldn’t be too surprised if someone who thinks about this more than me responded in a way that updated me considerably. But, I think I may be communicating a point that has reverse inferential distance – i.e the points I’m making are so obvious that they don’t seem relevant to the discussion, and my point is that if you’re not used to thinking in exponential terms they aren’t obvious, so this subthread may be most useful to people who happen to be feel confused or that things are unintuitive in the way I feel right now)
You could fit a line through two points, you could fit an exponential through two points, you could fit anything!
I mean, presumably there are more data points you could (at least hypothetically) have included, in which it’s not literally a single discontinuity, but a brief switch to a sharp increase in progress, followed by a return to something closer to the original curve. I’m not sure about the technical definition of discontinuity, but in a world where the graph had a point for each month instead of year, but the year of 2007 still had such a sharp uptick, the point doesn’t stop being interesting.
Since the Chess graph is uniquely confusing (hence my original confusion), I’d answer the rest of your question with, say, a more generic economic growth model.
If the economy were growing linearly, and then had a brief spike, and then returned to growing linearly at roughly the same rate, that’s one kind of interesting.
The fact that the economy grows exponential is a different kind of interesting, which layfolk routinely make bad choices due to poor intuitions about. (i.e. this is why investing is a much better idea that it seems, and why making tradeoffs that involve half-percent sacrifices to economic growth are a big deal. If you’re used to thinking about it this way it may not longer seem interesting, but, like, there are whole courses explaining this concept because it’s non-obvious)
If the economy is growing exponentially, and there’s a discontinuity where for one year it grows much more rapidly, that’s a third kind of interesting, and it’s in turn different interesting whether growth slows back down such that it seems like it’s at a similar rate to what we had before the spike, or continues as the spike had basically let you skip several years and then continue at an even faster rate.
Are you talking about the overall trend, or the discontinuity itself?
It doesn’t seem like it would make sense to talk about the discontinuity itself as linear / exponential, since it’s defined by just two points: the point before the jump, and the point after the jump. You could fit a line through two points, you could fit an exponential through two points, you could fit anything!
(If you had a trend that switched from being linear to being exponential, that would be a different story. But this graph doesn’t look like that to me.)
Have I misunderstood what you’re saying?
I was mostly talking about the overall trend, although I have additional thoughts on your point about the point-of-discontinuity.
(epistemic status: a bit outside my comfort zone. I feel confident but wouldn’t be too surprised if someone who thinks about this more than me responded in a way that updated me considerably. But, I think I may be communicating a point that has reverse inferential distance – i.e the points I’m making are so obvious that they don’t seem relevant to the discussion, and my point is that if you’re not used to thinking in exponential terms they aren’t obvious, so this subthread may be most useful to people who happen to be feel confused or that things are unintuitive in the way I feel right now)
I mean, presumably there are more data points you could (at least hypothetically) have included, in which it’s not literally a single discontinuity, but a brief switch to a sharp increase in progress, followed by a return to something closer to the original curve. I’m not sure about the technical definition of discontinuity, but in a world where the graph had a point for each month instead of year, but the year of 2007 still had such a sharp uptick, the point doesn’t stop being interesting.
Since the Chess graph is uniquely confusing (hence my original confusion), I’d answer the rest of your question with, say, a more generic economic growth model.
If the economy were growing linearly, and then had a brief spike, and then returned to growing linearly at roughly the same rate, that’s one kind of interesting.
The fact that the economy grows exponential is a different kind of interesting, which layfolk routinely make bad choices due to poor intuitions about. (i.e. this is why investing is a much better idea that it seems, and why making tradeoffs that involve half-percent sacrifices to economic growth are a big deal. If you’re used to thinking about it this way it may not longer seem interesting, but, like, there are whole courses explaining this concept because it’s non-obvious)
If the economy is growing exponentially, and there’s a discontinuity where for one year it grows much more rapidly, that’s a third kind of interesting, and it’s in turn different interesting whether growth slows back down such that it seems like it’s at a similar rate to what we had before the spike, or continues as the spike had basically let you skip several years and then continue at an even faster rate.