  We show that a rate of conditional Shannon entropy reduction, characterizing
the learning of an internal process about an external process, is bounded by
the thermodynamic entropy production. This approach allows for the definition
of an informational efficiency that can be used to study cellular information
processing. We analyze three models of increasing complexity inspired by the E.
coli sensory network, where the external process is an external ligand
concentration jumping between two values. We start with a simple model for
which ATP must be consumed so that a protein inside the cell can learn about
the external concentration. With a second model for a single receptor we show
that the rate at which the receptor learns about the external environment can
be nonzero even without any dissipation inside the cell since chemical work
done by the external process compensates for this learning rate. The third
model is more complete, also containing adaptation. For this model we show
inter alia that a bacterium in an environment that changes at a very slow
time-scale is quite inefficient, dissipating much more than it learns. Using
the concept of a coarse-grained learning rate, we show for the model with
adaptation that while the activity learns about the external signal the option
of changing the methylation level increases the concentration range for which
the learning rate is substantial.
