Accurate step-hold tracking of smoothly varying periodic and aperiodic probability

Stepwise updating of probabilities is neither universal nor fully explained by motor costs

Often our expectations are set by sampling many times from the same distribution. When that distribution changes, so should our expectations. If we want to decide whether to take our umbrella today we need to have tracked, and updated, our estimate of the probability of rain by reference to recent temperatures and precipitation. Under debate is whether we update our mental estimates of probabilities given each new incoming piece of evidence or whether we stick with a current estimate until it becomes clearly in need of change. Previous research has suggested that participant estimates of running probabilities are not updated on every trial, but only intermittently. This has been used to support change-point models of probability updating. However, such a pattern could also be explained by the way common laboratory procedures impose a motor cost to update the probability report. This study was designed to remove the motor confound. Our procedure required similar motor actions for both changing and maintaining one's probability estimate. At a group level, motor cost did affect updating frequency and removing the default response option encouraged more frequent updating. However, intermittent updating response patterns did not disappear completely. Despite this equivalence in response effort, some participants, even when forced to make a new estimate on every trial, continued to update rarely while other participants meticulously updated every trial. We conclude deliberate updating frequency is heterogenous but intermittent updating is not simply an artifact of motor cost.

Individual differences in the perception of probability

In recent studies of humans estimating non-stationary probabilities, estimates appear to be unbiased on average, across the full range of probability values to be estimated. This finding is surprising given that experiments measuring probability estimation in other contexts have often identified conservatism: individuals tend to overestimate low probability events and underestimate high probability events. In other contexts, repulsive biases have also been documented, with individuals producing judgments that tend toward extreme values instead. Using extensive data from a probability estimation task that produces unbiased performance on average, we find substantial biases at the individual level; we document the coexistence of both conservative and repulsive biases in the same experimental context. Individual biases persist despite extensive experience with the task, and are also correlated with other behavioral differences, such as individual variation in response speed and adjustment rates. We conclude that the rich computational demands of our task give rise to a variety of behavioral patterns, and that the apparent unbiasedness of the pooled data is an artifact of the aggregation of heterogeneous biases.

Finding numbers in the brain

After listing functional constraints on what numbers in the brain must do, I sketch the two's complement fixed-point representation of numbers because it has stood the test of time and because it illustrates the non-obvious ways in which an effective coding scheme may operate. I briefly consider its neurobiological implementation. It is easier to imagine its implementation at the cell-intrinsic molecular level, with thermodynamically stable, volumetrically minimal polynucleotides encoding the remembered numbers, than at the circuit level, with plastic synapses encoding them.This article is part of a discussion meeting issue 'The origins of numerical abilities'.

Forecasting the outcome of a time-varying Bernoulli process: Data from a laboratory experiment

The data presented in this article are related to the research article entitled "Discrete Adjustment to a Changing Environment: Experimental Evidence" (Khaw et al., 2017) [1]. We present data from a laboratory experiment that asks subjects to forecast the outcome of a time-varying Bernoulli process. On a computer program, subjects draw rings with replacement from a virtual box containing green and red rings in an unknown proportion. Subjects provide their estimates of the probability of drawing a green ring. They are rewarded for their participation and for the accuracy of their estimates. The actual probability of drawing a green ring is initially drawn from a uniform distribution. It then changes intermittently throughout the session, and each subsequent probability is an independent draw from the uniform distribution. Each session involves 1000 ring draws. The dataset contains the values of the underlying probability, the sequence of ring draws that are realized, and the subjects' estimates and response times. The dataset contains the performance of 11 subjects who each completed 10 sessions over the course of several days.

Theoretical implications of quantitative properties of interval timing and probability estimation in mouse and rat

In three experiments with mice ( Mus musculus ) and rats (Rattus norvigicus), we used a switch paradigm to measure quantitative properties of the interval-timing mechanism. We found that: 1) Rodents adjusted the precision of their timed switches in response to changes in the interval between the short and long feed latencies (the temporal goalposts). 2) The variability in the timing of the switch response was reduced or unchanged in the face of large trial-to-trial random variability in the short and long feed latencies. 3) The adjustment in the distribution of switch latencies in response to changes in the relative frequency of short and long trials was sensitive to the asymmetry in the Kullback-Leibler divergence. The three results suggest that durations are represented with adjustable precision, that they are timed by multiple timers, and that there is a trial-by-trial (episodic) record of feed latencies in memory.