Optimal mixtures of three space-time-magnitude earthquake likelihood models are found for forecasting earthquakes with magnitudes of 5.0 and greater in the New Zealand and California catalogs, with forecasting time horizons ranging from 0 to 3000 days. The models are the Epidemic-Type Aftershock (ETAS) short-term clustering model, the Every Earthquake a Precursor According to Scale (EEPAS) medium-term clustering model, and the Proximity to Past Earthquakes (PPE) quasi-time-invariant smoothed seismi city model. The ETAS model is by far the most informative of these models for short time horizons of a few days, but even with a zero time horizon, an optimal mixture of the three models, here called the Janus model, outperforms it with an information gain per earthquake (IGPE) of about 0.1. For time horizons of 10-3000 days, the Janus model outperforms the most informative of its component models with IGPEs ranging from 0.2 to 0.5. As the time horizon lengthens beyond six months in New Zealand and two years in California, the EEPAS model becomes the most informative of the individual models and the major component of the optimal mixture. Changes in the Janus model parameters with the forecasting time horizon reveal features of time-and-area scaling of precursory seismicity. The results suggest that both cascades of triggering and the precursory scale increase phenomenon contribute to earthquake predictability and that these contributions are largely independent.
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