  We present an unbinned likelihood analysis formalism employing photon weights
-- the probabilities that events are associated with a particular source. This
approach is applicable to any photon-resolving instrument, and thus well suited
to high-energy observations; we focus here on GeV $\gamma$-ray data from the
Fermi Large Area Telescope. Weights connect individual photons to the outputs
of a detailed, expensive likelihood analysis of a much larger data set. The
weighted events can be aggregated into arbitrary time spans ranging from
microseconds to years. Such retrospective grouping permits time- and
frequency-domain analysis over a wide range of scales and enables
characterization of disparate phenomena like blazar flares, $\gamma$-ray
bursts, pulsar pulses, novae, $\gamma$-ray binaries, and other variable
sources. To demonstrate the formalism, we incorporate photon weights into the
Bayesian blocks algorithm and perform a hierarchical time scale analysis of 3C
279 activity. We analyze pulsar pulse profiles and estimate the unpulsed
emission level and the optimal division of the data into on- and off-pulse
intervals. We extend the formalism to Fourier analysis and derive estimators
for power spectra, used to search for and characterize periodic sources. We
show how the Fast Fourier transform can be used to probe orbital periods as
short as a minute and we discuss mitigation of spurious signals. Our final
example combines time- and frequency- domain analysis to jointly characterize
the flares and orbital modulation of Cygnus X-3, yielding the strongest
detection of the orbital signal ($>$13$\sigma$) to date. Finally, we discuss
extensions of the work to other GeV sources and to X-ray and TeV observations.
