Hierarchical Models in the Brain
Figure 6
The predictions and conditional densities on the states and parameters of the linear convolution model of the previous figure.
Each row corresponds to a level, with causes on the left and hidden states on the right. In this case, the model has just two levels. The first (upper left) panel shows the predicted response and the error on this response (their sum corresponds to the observed data). For the hidden states (upper right) and causes (lower left) the conditional mode is depicted by a coloured line and the 90% conditional confidence intervals by the grey area. These are sometimes referred to as “tubes”. Finally, the grey lines depict the true values used to generate the response. Here, we estimated the hyperparameters, parameters and the states. This is an example of triple estimation, where we are trying to infer the states of the system as well as the parameters governing its causal architecture. The hyperparameters correspond to the precision of random fluctuations in the response and the hidden states. The free parameters correspond to a single parameter from the state equation and one from the observer equation that govern the dynamics of the hidden states and response, respectively. It can be seen that the true value of the causal state lies within the 90% confidence interval and that we could infer with substantial confidence that the cause was non-zero, when it occurs. Similarly, the true parameter values lie within fairly tight confidence intervals (red bars in the lower right).
