In the multivariate-normal case, the two approaches are exactly equivalent: if you know the marginals (unconditional true effectiveness and unconditional estimate value), and R^2, then you know the entire shape of the distribution (and hence the distribution of the true mean given the estimated mean).
A model in which the estimate is bivariate normal with R^2=0.9 to the ground truth corresponds to an estimate distribution of, if my stats is right, X~N(0, 0.9), E~N(0, 0.1), Y=X+E (where X is the ground truth, E the error, and Y the estimate; the second arguments are variances; this is true up to an affine transformation). As such, it follows from e.g. this theorem cited on Wikipedia that the actual mean scales linearly with the measured mean, although the coefficient of correlation is not quite what Gregory said (it’s R, not R^2).
In the multivariate-normal case, the two approaches are exactly equivalent: if you know the marginals (unconditional true effectiveness and unconditional estimate value), and R^2, then you know the entire shape of the distribution (and hence the distribution of the true mean given the estimated mean).
A model in which the estimate is bivariate normal with R^2=0.9 to the ground truth corresponds to an estimate distribution of, if my stats is right, X~N(0, 0.9), E~N(0, 0.1), Y=X+E (where X is the ground truth, E the error, and Y the estimate; the second arguments are variances; this is true up to an affine transformation). As such, it follows from e.g. this theorem cited on Wikipedia that the actual mean scales linearly with the measured mean, although the coefficient of correlation is not quite what Gregory said (it’s R, not R^2).
Thanks Ben, you’re exactly right. I’d convinced myself of the contrary with a spurious geometric argument.