Our previous study proposed a Bayesian framework to enhance the approach by the renewal process to forecast earthquakes' timing and tested it in simulated seismicity-like time series. As a first step toward applying the Bayesian approach to actual seismic activity, it is crucial to use seismic catalogs for examination of the probability density functions in Bayes' theorem, which is in its simplest form in the Bayesian approach: the inter-event time distribution and the conditional and inverse probability between inter-event times at two cutoff magnitudes. In this study, I examined the properties of these probability density functions using time series with weak inter-event correlations extracted from three seismic catalogs: stationary time series with a nearly constant occurrence rate and aftershock sequences transformed by the Omori Utsu law. I found a new scaling property related to the temporal hierarchy of seismic activity. Using this property, I derived the above three probability density functions. Regarding the inter-event time distribution, I discuss its approximate scaling universality from the viewpoint of temporal fluctuations of transformed seismic time series by instantaneous occurrence rate. The derived inverse probability enables probabilistic evaluation of the large earthquakes' timing in the simplest Bayesian approach. Finally, I discuss extending it to the general Bayesian approach toward its practical use.