Cosmological ensemble and directional averages of observables
  We show that at second order, ensemble averages of observables and
directional averages do not commute due to gravitational lensing -- observing
the same thing in many directions over the sky is not the same as taking an
ensemble average. In principle this non-commutativity is significant for a
variety of quantities that we often use as observables and can lead to a bias
in parameter estimation. We derive the relation between the ensemble average
and the directional average of an observable, at second order in perturbation
theory. We discuss the relevance of these two types of averages for making
predictions of cosmological observables, focusing on observables related to
distances and magnitudes. In particular, we show that the ensemble average of
the distance in a given observed direction is increased by gravitational
lensing, whereas the directional average of the distance is decreased. For a
generic observable, there exists a particular function of the observable that
is not affected by second-order lensing perturbations. We also show that
standard areas have an advantage over standard rulers, and we discuss the
subtleties involved in averaging in the case of supernova observations.
