  Motivated by recent progress in the statistical modeling of quasar
variability, we develop a new approach to measuring emission-line reverberation
lags to estimate the size of broad-line regions (BLRs) in active galactic
nuclei. Assuming that all emission-line light curves are scaled, smoothed, and
displaced versions of the continuum, this alternative approach fits the light
curves directly using a damped random walk model and aligns them to recover the
time lag and its statistical confidence limits. We introduce the mathematical
formalism of this approach and demonstrate its ability to cope with some of the
problems for traditional methods, such as irregular sampling, correlated
errors, and seasonal gaps. We redetermine the lags for 87 emission lines in 31
quasars and reassess the BLR size--luminosity relationship using 60 H-beta
lags. We confirm the general results from the traditional cross-correlation
methods, with a few exceptions. Our method, however, also supports a broad
range of extensions. In particular, it can simultaneously fit multiple lines
and continuum light curves which improves the lag estimate for the lines and
provides estimates of the error correlations between them. Determining these
correlations is of particular importance for interpreting emission-line
velocity--delay maps. We can also include parameters for luminosity-dependent
lags or line responses. We use this to detect the scaling of the BLR size with
continuum luminosity in NGC 5548.
