¹⁹The meaning of “weaker” signalling is a bit vague in this intuitive example. To make it more rigorous, we can consider, as an illustrative example, one of the simplest online learning tasks, namely linear regression. There, we minimize a quantity such as ∑ _t(y_t - ax_t)²∕σ _t² where x is input, y is output, σ _t² is noise level, and a is a parameter to be estimated. The magnitude of the error signal for each data point is proportional to the inverse of the noise level σ_t². Thus, for a high noise level (large uncertainty), the error signal is smaller. If the noise level is estimated separately for each data point (or time point t), this will have the effect of reducing the error signal at time points where there is a lot of uncertainty as modelled by the noise level σ_t². The concrete algorithm used here might be what is called the delta rule; see Korenberg and Ghahramani (2002) as an example of a related if slightly more complex model. Mai et al. (2022) propose a closely related weighting in the context of reinforcement learning and RPE. See also footnote 15 in Chapter 5.