We present a Bayesian updating method on the inter-event times at
different magnitude thresholds in a marked point process, toward the
probabilistic forecasting of an upcoming large event using temporal
information on smaller events. Bayes' theorem in a marked point process
that yields the one-to-one relationship between intervals at lower and
upper magnitude thresholds is presented. This theorem is extended to
Bayesian updating for an uncorrelated marked point process that yields
the relationship between multiple consecutive lower intervals and one
upper interval. The inverse probability density function and its
approximation function are derived. For the former, the condition for
having a peak is shown. The latter is easier to apply to the time series
of the ETAS model, and it consists of the kernel part, which includes
the product of the conditional probabilities, and the correction term.
The maximum point of the kernel part is shown to be not significantly
affected by the correction term when applying the Bayesian updating to
the ETAS model time series numerically. The occurrence time of the
upcoming large event is estimated using this maximum point, and its
accuracy is evaluated considering the relative error with the actual
occurrence time. Moreover, forecasting is evaluated to be effective by
the continuity of the updates with the accuracy within an acceptable
range prior to the upcoming large event. Under these conditions, the
statistical analysis indicates that forecasting is relatively effective
immediately or long after the last major event in which stationarity is
dominant in the time series.