  Photon loss rates set an effective upper limit on the size of computations
that can be run on current linear optical quantum devices. We present a family
of techniques to mitigate the effects of photon loss on both output
probabilities and expectation values derived from noisy linear optical circuits
composed of an input of $n$ photons, an $m$-mode interferometer, and $m$ single
photon detectors. Central to these techniques is the construction of objects
called recycled probabilities. Recycled probabilities are constructed from
output statistics affected by loss, and are designed to amplify the signal of
the ideal (lossless) probabilities. Classical postprocessing techniques then
take recycled probabilities as input and output a set of loss-mitigated
probabilities, or expectation values. We provide analytical and numerical
evidence that these methods can be applied, up to large sample sizes, to
produce more accurate outputs than those obtained from postselection - which is
currently the standard method of coping with photon loss when sampling from
discrete variable linear optical quantum circuits. In contrast, we provide
strong evidence that the popular zero noise extrapolation technique cannot
improve on on the performance of postselection for any photon loss rate.
