For the second case, each data point can measure something different, possibly correlated with each other, and related in different ways to the parameters we’re trying to estimate. For instance, maybe we’re trying to estimate some parameters of a car, so we measure the wheel sizes, axle length, number of gears, engine cylinder volume, etc, etc. Every now and then we measure something which gives us a totally different “kind” of information from the other things we measured—something which forces a non-exponential-family update. When that happens, we have to add a new summary component. Eventually other data points may measure the same “kind” of information and also contribute to that component of the summary. But over time, it becomes more and more rare to measure something which no other data point has captured before, so we add summary dimensions more and more slowly.
(Side note: there’s no reason I know why O(log n) growth would be special here; the qualitative story would be similar for any sub-linear summary growth.)
For the second case, each data point can measure something different, possibly correlated with each other, and related in different ways to the parameters we’re trying to estimate. For instance, maybe we’re trying to estimate some parameters of a car, so we measure the wheel sizes, axle length, number of gears, engine cylinder volume, etc, etc. Every now and then we measure something which gives us a totally different “kind” of information from the other things we measured—something which forces a non-exponential-family update. When that happens, we have to add a new summary component. Eventually other data points may measure the same “kind” of information and also contribute to that component of the summary. But over time, it becomes more and more rare to measure something which no other data point has captured before, so we add summary dimensions more and more slowly.
(Side note: there’s no reason I know why O(log n) growth would be special here; the qualitative story would be similar for any sub-linear summary growth.)