Fig 1.
This is a form of graphical notation that expresses the conditional dependencies in the generative model. Random variables are shown in circles, with arrows indicating causal influences. The progression from one state (sτ) to the next (sτ+1) is affected by the precision (ω) of the transition matrix. This stochasticity results in randomness in the observed outcomes (oτ). The panel below specifies the parameterisation of this model. Notably, in addition to (likelihood) beliefs about how states generate data (A), and beliefs about state transitions (B), we need prior beliefs about the precision of these transitions. These take the form of a gamma distribution, P(ω) ∝ βe−β·ω, where the current prior belief (ω) is a function of the most recent posterior beliefs (β). This has the convenient property that the expectation of the prior beliefs as we update them are the value of the most recent posterior beliefs [22].
Fig 2.
Sequences of outcomes emerging from random transition dynamics.
Fig 2a shows the sequence of outcomes (o) produced by the sequence of states (s) when the transition matrices for both sequences are deterministic. Fig 2b shows the sequence of states, and outcomes those states generate, when the matrix defining the transitions between states (B) is not deterministic. Even though the transitions from states to outcomes are still deterministic, as in Fig 2a, the final sequence of outcomes is different. In Fig 2c we show the consequences of introducing randomness into the transitions (matrix A) from states to events. The sequence of states remains the same as in Fig 2a, but the outcomes generated by each state are random, resulting in a different sequence of outcomes. Figs 2a and 2c demonstrate how randomness in the transition matrices can results in unpredictable sequences of outcomes.
Fig 3.
B matrices created for the generative process.
Lighter values indicate probabilities closer to 1. As the precision decreases, shown in Figs 3a through 3d, the probability of a number-absence state transitioning to the next number decreases, while the probability of a transition to the preceding number increases. The grey transparent box covers transitions into and within the final 1s break. Separate states are required for each of the number-absence states to ensure this matrix generates a reliable sequence of numbers. Number-absence states following 5,6,7 may transition to the first of the block of 4 number-absence states as the precision decreases, to ensure that the sequence of numbers is always 8 long. These matrices can result in unexpected sequences, as shown in Fig 3e, which is drawn from the matrix defining a scenario with precision = 0.39, since it only has one error.
Fig 4.
Sequences and their volatility.
25 sequences are presented in each block. The first sequence is 1–8. The following 24 sequences are placed into 8 sets; each set has a predetermined volatility; alternatively, the y-axis gives the volatility of the generative process. The exact sequences are shown hanging below the x-axis in Fig 4a, where unexpected numbers resulting from an aberrant transition are highlighted in bold font. ω-1 is used here to distinguish the volatility of the generative process from the participants’ inferred volatility β. Fig 4b shows the composition of combination sequences. The same set of 25 sequences is taken, in the same order, to generate the combination sequences. Two numbers in each sequence are exchanged for different numbers to generate a new sequence. The upper row shows sequences with varying degrees of volatility (but precise likelihood probabilities), while the lower row shows these same sequences augmented as if they were generated from imprecise likelihood distributions.
Fig 5.
This figure shows the data when averaged across all blocks and participants. There are 25 oscillations, which correspond to the 25 sequences, and we explain this in terms of an optical effect. In the next section, we propose that the slow fluctuations that characterise this data are explained best by inferred precision. The grand mean data is shown in green, while the data with the fluctuations regressed out is shown in purple. A trace indicating the precision of the sequences is overlaid—referred to as the adjusted precision, to differentiate from the inferred precision of the participants.
Fig 6.
Simulations of the inferred precision.
Three simulations are generated with prior precisions of 1, 1.5 and 3.25, in blue, green and red respectively (see Eq 2 for a formal definition of prior precision).
Table 1.
Model designs.
Fig 7.
Design matrices for models 1–6, corresponding to the stimuli detailed in Table 1 and the additional two naïve models (models 5 and 6). Each column indicates a single regressor, while each row is a point in time. In each model the final column (un-numbered) is a constant factor that accounts for the z-scoring performed in the pre-processing. The scale bar on the right indicates the values of the cells in the matrix.
Fig 8.
Bayesian comparison of alternative models relative to the null model.
Fig 8a shows a bar plot of the log model evidence relative to that of the null model at a prior precision of 1, with the R2 scores for each fit provided above each bar (the lowest of our candidates, see Fig 9a). Fig 8b shows the posterior probabilities of the models, calculated by passing the log model evidence through a softmax function. In the case of Fig 8b, we take the model comparison conducted at a prior precision of β-1 = 2.25, to justify the model selected in the next section. Fig 8c shows the log model evidence for models 2–6 relative to that of model 1 (the null model), plotted against the prior belief over precision used to generate the design matrix for each mode, and can be thought of as a series of individual model comparisons. The curve for model 3 is in bold to indicate this is the model used for the analysis of participant’s prior beliefs. Note that the null model and models 5–6 do not depend on inferred precision and are therefore invariant to the prior beliefs. These results show that models explicitly containing inferred precision perform better over the range of prior beliefs considered, with the simpler model (model 3) performing best for β-1 > 2.25.
Fig 9.
Estimating participants prior beliefs over precision.
Using a Bayesian model comparison, we can calculate a posterior distribution over prior beliefs about precision using the evidence for models of observed data, under each level of prior precisions considered. In Fig 9a, posterior distributions for individual participants are shown in blue for 5 of 9, while the remaining 4 are given colours that correspond to their models in parts c-f. The insert for Fig 9a shows the results for two participants when we analyse the data for the different sequences separately, with the red curves indicating the estimated prior precision for the sequences with imprecise A and B matrices, and the black showing the estimated prior precision for sequences with only an imprecise B matrix. The blue lines replicate the data in the main part of Fig 9a, with the two sequences combined. In Fig 9b we show a ‘confusion’ matrix, where the elements show the probability that the prior precision represented by the column was the value used to generate the simulated data represented in each row. Figs 9c-f show the simulated data generated with the most likely prior precision, in the colours given in Fig 9a, overlaid on the recorded data (in black). The estimation of the participant’s inferred precision is also provided in pink, to show the contribution of precision to the model fit, as well as to show the fine scale tracking and responsive of pupil diameter to changes in environmental volatility. The four participants shown had the highest posterior probability in the estimation of their prior beliefs over precision. R2 values are given for the model fits for the 4 participants shown.