The extent to which language processing involves prediction of upcoming inputs remains a question of ongoing debate. One important data point comes from DeLong et al. (2005) who reported that an N400-like event-related potential correlated with a probabilistic index of upcoming input. This result is often cited as evidence for gradient probabilistic prediction of form and/or semantics, prior to the bottom-up input becoming available. However, a recent multi-lab study reports a failure to find these effects (Nieuwland et al., 2017). We review the evidence from both studies, including differences in the design and analysis approach between them. Building on over a decade of research on prediction since DeLong et al. (2005)'s original study, we also begin to spell out the computational nature of predictive processes that one might expect to correlate with ERPs that are evoked by a functional element whose form is dependent on an upcoming predicted word. For paradigms with this type of design, we propose an index of anticipatory processing, Bayesian surprise, and apply it to the updating of semantic predictions. We motivate this index both theoretically and empirically. We show that, for studies of the type discussed here, Bayesian surprise can be closely approximated by another, more easily estimated information theoretic index, the surprisal (or Shannon information) of the input. We re-analyze the data from Nieuwland and colleagues using surprisal rather than raw probabilities as an index of prediction. We find that surprisal is gradiently correlated with the amplitude of the N400, even in the data shared by Nieuwland and colleagues. Taken together, our review suggests that the evidence from both studies is compatible with anticipatory semantic processing. We do, however, emphasize the need for future studies to further clarify the nature and degree of form prediction, as well as its neural signatures, during language comprehension.