Extended Data Fig. 2 |. Validation of the COIN model. a, Validation of the inference algorithm of the COIN model with a single context.
We computed inferences in the COIN model with a single context based on synthetic observations (state feedback) generated by its generative model (Fig. 1a). Plots show the cumulative distributions of posterior predictive p-values of the state variable (left), and the parameters governing its dynamics (retention, middle; drift, right). The posterior predictive p-value is computed by evaluating the cumulative distribution function of the model’s posterior over the given quantity at the true value of that quantity (as defined by the generative model). Empirical distributions of posterior predictive p-value were collected across 4000 simulations (with different true state dynamics parameters), with 500 time steps in each simulation (during which the true state changes, but the state dynamics parameters are constant). Note that although true state dynamics parameters do not change during a simulation, inferences in the model about them will still generally evolve, and so a new posterior p-value is generated in each time step even for these quantities. If the model implements well-calibrated probabilistic inference under the correct generative model, all these empirical distributions should be uniform. This is confirmed by all cumulative distributions (orange and purple curves) approximating the identity line (black diagonal). Orange curves show posterior predictive p-values under the corresponding marginals of the model’s posterior. To give additional information about the model’s joint posterior over state dynamics parameters, we also show the posterior predictive p-value (cumulative) distribution of each parameter conditioned on the true value of the other one (purple curves). b, Validation of the inference algorithm of the COIN model with multiple contexts. Simulations as in a but with additional synthetic observations (sensory cues) and multiple contexts allowed both during data generation and inference. Empirical distributions of posterior predictive p-value were collected across 2000 simulations (with different true retention and drift parameters), with 500 time steps in each simulation (during which not only states evolve but also contexts transition, and sometimes novel contexts are created). Left column shows the true distributions of sensory cues, contexts and parameters. Inset shows the growth of the number of contexts over time both during generation (blue) and inference (orange). Middle and right columns show the cumulative probabilities of the posterior predictive p-values (pooled across data sets and time steps) for the observations (top row), contexts and state (middle row) and parameters (bottom row). To calculate the posterior predictive p-values for the context, inferred contexts were relabelled by minimising the Hamming distance between the relabelled context sequence and the true context sequence (see Suppl. Inf.). For the parameters, the posterior predictive p-values were calculated with respect to both the marginal distributions (retention and drift) and the conditional distributions (retention | drift, and drift | retention) as in a. The cumulative probability curves approximate the identity line (thin black line) showing that the inferred posterior probability distributions are well calibrated. c, Parameter recovery in the COIN model related to Fig. 2. Plots show the COIN model parameters that were recovered (y-axes) from fits to 10 synthetic data sets generated with the COIN model parameters (true, x-axes) obtained from the fits to each participant in the spontaneous (n = 8) and evoked (n = 8) recovery experiments (Extended Data Fig. 3). Vertical bars show the interquartile range of the recovered parameters for each participant. While several parameters are recovered with good accuracy (σq, μa, σd, σm), others are not (α, and in particular σa and ρ). We expect that with richer paradigms and larger data sets, all parameters would be recovered accurately. Most importantly, despite partial success with recovering individual parameters, model recovery shows that recovered parameter sets taken as a whole can be used to accurately identify whether data was generated by the dual-rate or COIN model (d). Note that we make no claims about individual parameters in this study as our focus is on model class recovery. d-e, Model recovery for spontaneous (d) and evoked recovery experiments (e) related to Fig. 2. Synthetic data sets were generated using one of two models (COIN model, red; dual-rate model, blue). Parameters used for each model were those obtained from the fits to each participant in the spontaneous (n = 8) and evoked (n = 8) recovery experiments (Extended Data Fig. 3) – i.e. for the COIN model, these were the same synthetic data sets as those used in c. Then, the same model comparison method that we used on real data (Fig. 2c, e, insets) was used to recover the model that generated each synthetic data set (see Methods). Arrows connect true models (used to generate synthetic data, disks on top) to models that were recovered from their synthetic data (pie-chart disks at bottom). Arrow colour indicates identity of recovered model, arrow thickness and percentages indicate probability of recovered model given true model. Bottom disk sizes and pie-chart proportions show total probability of recovered model and posterior probability of true model given recovered model (assuming a uniform prior over true models), respectively, with percentages specifically indicating posterior probability of the correct model. These results show that the model recovery process is generally very accurate and actually biased against the COIN model in favour of the dual-rate model.