  We propose two methods to calibrate the parameters of the epidemic-type
aftershock sequence (ETAS) model based on expectation maximization (EM) while
accounting for temporal variation of catalog completeness. The first method
allows for model calibration on long-term earthquake catalogs with temporal
variation of the completeness magnitude, $m_c$. This calibration technique is
beneficial for long-term probabilistic seismic hazard assessment (PSHA), which
is often based on a mixture of instrumental and historical catalogs. The second
method generalizes the concept of $m_c$, considering rate- and
magnitude-dependent detection probability, and allows for self-consistent
estimation of ETAS parameters and high-frequency detection incompleteness. With
this approach, we aim to address the potential biases in parameter calibration
due to short-term aftershock incompleteness, embracing incompleteness instead
of avoiding it. Using synthetic tests, we show that both methods can accurately
invert the parameters of simulated catalogs. We then use them to estimate ETAS
parameters for California using the earthquake catalog since 1932. To explore
how model calibration, inclusion of small events, and accounting for short-term
incompleteness affect earthquakes' predictability, we systematically compare
variants of ETAS models based on the second approach in pseudo-prospective
forecasting experiments for California. Our proposed model significantly
outperforms the ETAS null model, with decreasing information gain for
increasing target magnitude threshold. We find that the ability to include
small earthquakes for simulation of future scenarios is the primary driver of
the improvement and that accounting for incompleteness is necessary. Our
results have significant implications for our understanding of earthquake
interaction mechanisms and the future of seismicity forecasting.
