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Optimal prediction with resource constraints using the information bottleneck
Fig 3
We consider the task of predicting the path of an SDDHO with 
and Δt = 1.
(a) (left) We encode the history of the stimulus, Xt, with a representation generated by the information bottleneck, 
, that can store 1 bit of information. Knowledge of the coordinates in the compressed representation space enables us reduce our uncertainty about the bar’s position and velocity, with a confidence interval given by ellipse in yellow. This particular choice of encoding scheme enables us to predict the future, Xt+Δt with a confidence interval given by the purple ellipse. The information bottleneck guarantees this uncertainty in future prediction is minimal for a given level of encoding. (right) The uncertainty in the prediction of the future can be reduced by reducing the overall level of uncertainty in the encoding of the history, as demonstrated by increasing the amount of information 
can store about Xt. However, the uncertainty in the future prediction cannot be reduced below the variance of the propagator function. (b) We show how the information with Xt+Δtscales with the information about Xt, highlighting the points represented in panel A.
doi: https://doi.org/10.1371/journal.pcbi.1008743.g003