The section on forecasting quantities is really a special case of estimating any unknown quantity X given a prior and data when X has a true (but unknown) value different from the prior. You should expect your updates to be roughly monotonic from your prior value to the true value as data accumulates.
For example, if X is vaccine effectiveness of an unusually effective vaccine then the expected value of your prior for X is going to be too small. As studies accumulate, they should all point to the true value of X (up to errors/biases/etc) and your posterior mean should also move towards the true value.
An important difference here is foresight vs hindsight. At any particular time, you expect your future posterior mean of X to be your current posterior mean of X. However, once you know what your final posterior mean of X to be (approximately the true value) then updates up to that point look .
The section on forecasting quantities is really a special case of estimating any unknown quantity X given a prior and data when X has a true (but unknown) value different from the prior. You should expect your updates to be roughly monotonic from your prior value to the true value as data accumulates.
For example, if X is vaccine effectiveness of an unusually effective vaccine then the expected value of your prior for X is going to be too small. As studies accumulate, they should all point to the true value of X (up to errors/biases/etc) and your posterior mean should also move towards the true value.
An important difference here is foresight vs hindsight. At any particular time, you expect your future posterior mean of X to be your current posterior mean of X. However, once you know what your final posterior mean of X to be (approximately the true value) then updates up to that point look .
Appendix: maths proving the above for a simple case.