  Sensors often serve at least two purposes: predicting their input and
minimizing dissipated heat. However, determining whether or not a particular
sensor is evolved or designed to be accurate and efficient is difficult. This
arises partly from the functional constraints being at cross purposes and
partly since quantifying the predictive performance of even in silico sensors
can require prohibitively long simulations. To circumvent these difficulties,
we develop expressions for the predictive accuracy and thermodynamic costs of
the broad class of conditionally Markovian sensors subject to unifilar hidden
semi-Markov (memoryful) environmental inputs. Predictive metrics include the
instantaneous memory and the mutual information between present sensor state
and input future, while dissipative metrics include power consumption and the
nonpredictive information rate. Success in deriving these formulae relies
heavily on identifying the environment's causal states, the input's minimal
sufficient statistics for prediction. Using these formulae, we study the
simplest nontrivial biological sensor model---that of a Hill molecule,
characterized by the number of ligands that bind simultaneously, the sensor's
cooperativity. When energetic rewards are proportional to total predictable
information, the closest cooperativity that optimizes the total energy budget
generally depends on the environment's past hysteretically. In this way, the
sensor gains robustness to environmental fluctuations. Given the simplicity of
the Hill molecule, such hysteresis will likely be found in more complex
predictive sensors as well. That is, adaptations that only locally optimize
biochemical parameters for prediction and dissipation can lead to sensors that
"remember" the past environment.
