How Do We Update Our Probability Estimates: Trial-by trial or all at once?

People have been wondering about this for decades. The direct Bayesian idea would be that after every observation your mental model of the relevant parameters for whatever stochastic event you are interested in gets revised to reflect your most recent evidence. However, in the lab that doesn't seem to be what people do.

I first became aware of this with Gallistel and colleagues excellent historical review and updated empirical work.¹ They argued instead for a "delta rule" where you have some model estimate and you use it until it is no longer good enough; then you refit your model and update your estimates. This step-wise updating idea came under scrutiny by Forsgren and colleagues. We first saw this as a pre-print ², but the full, revised version of their work was published about a year ago.³ They advocate for a more continuous updating, and also raise issues of procedural confounds. Julia Schirmeister, a PhD candidate at the University of Waterloo, took on evaluating one of these potential confounds in her master's work. Subsequent analyses, driven in part by excellent reviewer comments, have led to the recent acceptance of her work at Psychonomic Bulletin & Review. For such a brief report it is dense with interesting data ⁴ and insights. The main conclusion is that many participants do exhibit step-wise updating and that this cannot be explained solely by the motor confound that past tasks had built in: requiring more effort to update a reported probability estimate than to leave it unchanged. However, almost of equal interest is the individual heterogeneity that cannot be explained by simple task demands. Some people update essentially every trial, some rarely; even though, depending on context, this can take much more effort and time. If you have thoughts or questions please don't hesitate to reach out to Julia or me.