Methods

All calibration and experimental procedures were approved by the University of St Andrews Teaching and Research Ethics Committee.All participants gave informed consent.

Stimuli design and presentation.

In all experimental and calibration procedures visual stimuli were created in MATLAB 2015b (The Mathworks Inc) and presented using PsychToolbox (Brainard, 2007).

3.3.1 Proximal variance calibration experiment.

In order to provide an estimate of proximal variance that will allow the calculation of participant estimated Kalman gains we first undertake a proximal variance calibration experiment (see 3.8.2 page 64 for how we calculate proximal variance and relate it to our serial dependence experiments).

Participants

A total of eight participants undertook the proximal variance calibration experiment which followed a two alternative forced choice paradigm. (6 females, mean age 22, range 19-41). All participants were volunteers and recruited from the St Andrews SONA recruitment database.

Stimulus design and procedure

The proximal variance calibration experiment followed a two alternative forced choice paradigm in which participants were presented with an orientation discrimination task. Stimuli were oriented Gabor patches presented at either 5 or 20% contrast. All Gabors were embedded in Gaussian white noise (SD=15.5 cd/m2_{) Noise following Gabors is also Gaussian white noise and covered the whole} screen. All Gabor patches had a radius of 8 visual degrees and had a spatial frequency 0.5 cycles per visual degree. In trials pairs of Gabor stimuli were presented one after another in which the second Gabor was presented at a series of 7 different orientations from the first Gabor in the pair. These were +/- 1.5°, +/- 3.6°, +/- 5.7°, +/- 7.9°, +/- .10.1°, +/-12.16° and +/- 14.3. Each block presented 5 trials at both 5% and 20% contrast at each orientation difference giving 70 trials per block with each

participant completing 3 blocks thus performing 210 trials in total. Fixations were positioned centrally (see Fig 1 below). The procedure of the experiment in outlined in figure 1 below.

Figure 1. Proximal variance calibration experimental procedure. Participants were seated 57 cm from a CRT monitor. Each trial began with the presentation of a fixation cross in the centre of the screen for 250ms. Then a randomly oriented Gabor patch was presented for 500ms at 5% or 20% contrast depending on the condition then a noise mask for 500ms. Next a second Gabor was presented at one of 7 different angles ranging from (+/- 1.5°, +/- 3.6°, +/- 5.7°, +/- 7.9°, +/- 10.1°, +/-12.16° and +/- 14.3°) anti clock wise or clock wise from the first Gabor and then a second noise mask for 500ms. The task of the participant was to fixate on the fixation cross and then discriminate whether the orientation of the second Gabor was clock wise or anti clock wise of the first Gabor orientation. This was signalled by pressing j for clock wise and f for anti-clock wise. Each trial took approximately 6-8 seconds depending on how the response of the participants.After making a response, there was a 2-s delay during which only the fixation point was present before the onset of the next trial.

3.3.2 Main experiment one. Testing serial dependence under conditions of high distal

variance versus low and high proximal variance.

Experiment one aimed to produce a high level of distal variance and a range of proximal variance conditions and to assess serial dependence under such conditions.

Participants.

Main experiment one had a total of 10 participants (eight females, mean age 23, range 19-41) Participants were a different set to those who had completed the proximal variance calibration experiment.

Stimulus design and procedure

To produce a high level of distal variance, Gabor orientations were presented in a fully random sequence between 0 & 360°. We manipulated proximal variance, based in part on our proximal variance calibration and presented Gabors in the 5% and 20% contrasts used in this calibration experiment but also presented Gabors at 10% contrast as an exploratory measure. However, upon analysing the basic error variances for the 10% contrast condition we did not detect any differences between the 10% and 20% contrast conditions so do include data for this condition in our analyses and modelling. Gabor patches had a radius of 8 visual degrees and had a spatial frequency 0.5 cycles per visual degree. All Gabors were embedded in Gaussian white noise (SD=15.5 cd/m2). Each participant completed 3 blocks of trials. Each block comprised 210 trials comprised of 70 trials in each contrast condition (5%, 10% & 20%). This meant each participant completed 210 trials in each contrast condition and 630 trials in total. Condition order presentation was randomized across

participants. Blocks held the method of orientation change constant. The procedure and timings of the experiment are shown below in figure 2.

Figure 2. Stimulus design and procedure. Here we show how the experiment ran over two individual trials. Each trial began with the presentation of a blank screen (inter trial interval) for 250ms. Next a Gabor patch was shown in the centre of the screen for 500 ms, then a noise patch was presented for 500ms then an adjustment response bar was presented. The task of the participant was to move the adjustment response bar to try and match the orientation of the Gabor they had just

observed. Each trial lasted approximately 6-8 seconds. After making a response, there was a 250 ms delay during which only a blank screen was present before the onset of the next trial.

3.3.2 Main experiment two. Testing serial dependence under low distal variance versus low and high proximal variance.

Experiment two, aimed to produce a lower level of distal variance than in experiment 1 and two levels (high and low) of proximal variance conditions and to assess the magnitude of serial dependence under such conditions.

Participants.

A total of 9 participants took part in experiment two (seven females, mean age 24, range 19-41). Participants were a different group to experiment one and the proximal variance calibration experiment.All participants were recruited from St Andrews SONA recruitment database.

Stimulus design and procedure

Stimuli design, timings and procedure were identical to experiment one with the only difference being the variability of Gabor orientations. To reduce the distal variance of our Gabor stimulus orientations were presented in a sequence that followed a Gaussian random walk (μ=0, SD=11.552), (see below for details and an explanation of the reason for this). As the 10% contrast condition recorded no

differences in proximal variance to the 5% or 20% contrast conditions in experiment one, we did not include the 10% contrast condition in experiment two. Each participant completed 3 blocks. Each

block comprised 210 trials of 105 trials in each contrast condition. This meant each participant completed 315 trials in each contrast condition and 630 trials in total. Contrast condition order presentation was randomized across participants across blocks. All blocks kept the method of orientation change constant.

The reason we use a Gaussian random walk in our presentation of orientations in experiment two.

The term random walk, describes a stochastic process that follows a series of steps in some

mathematical space. In a Gaussian random walk, steps are drawn from a normal distribution with the variability of values from the mean of values determined by the standard deviation of the Gaussian distribution. In terms understanding the use of a Gaussian random walk in the current experiment, which has a mean of zero, it simply means that the next orientation value can be considered to be the same as the previous with some level of variance, which here is a standard deviation of 11.552. This has the twin effect of introducing some level of predictability in the sequence of orientations and as how far from the mean orientations are presented is constrained by the standard deviation of the random walk built into the stimulus design code, the overall level of variability in orientations over time is reduced.

3.4 Analyses and statistical tests