  We present a new formalism designed to discover dark matter annihilation
occurring in the Milky Way's dwarf galaxies. The statistical framework extracts
all available information in the data by simultaneously combining observations
of all the dwarf galaxies and incorporating the impact of particle physics
properties, the distribution of dark matter in the dwarfs, and the detector
response. The method performs maximally powerful frequentist searches and
produces confidence limits on particle physics parameters. Probability
distributions of test statistics under various hypotheses are constructed
exactly, without relying on large sample approximations. The derived limits
have proper coverage by construction and claims of detection are not biased by
imperfect background modeling. We implement this formalism using data from the
Fermi Gamma-ray Space Telescope to search for an annihilation signal in the
complete sample of Milky Way dwarfs whose dark matter distributions can be
reliably determined. We find that the observed data is consistent with
background for each of the dwarf galaxies individually as well as in a joint
analysis. The strongest constraints are at small dark matter particle masses.
Taking the median of the systematic uncertainty in dwarf density profiles, the
cross section upper limits are below the pure s-wave weak scale relic abundance
value (2.2 x 10^-26 cm^3/s) for dark matter masses below 26 GeV (for
annihilation into b quarks), 29 GeV (tau leptons), 35 GeV (up, down, strange,
and charm quarks and gluons), 6 GeV (electrons/positrons), and 114 GeV
(two-photon final state). For dark matter particle masses less than 1 TeV,
these represent the strongest limits obtained to date using dwarf galaxies.
