Table 4.

Induced prior distribution on root age ($t_{mrca}$) and age of Hymenoptera

Tree Prior	Prior Root Age	Prior Age of Hymenoptera (Ma)	Proportion of Ancestral Fossils
Uniform	486.6 (395.9, 650.3)	482.4 (392.2, 648.5)	—
$d\sim$ Exp(1)
FBD (Rnd)	1235 (920.1, 1532.4)	1194 (893.6, 1476.9)	0.00 (0.0, 0.009)
FBD (Div)	321.6 (315.0, 343.3)	265.1 (222.8, 316.2)	0.00 (0.00, 0.01)
pcFBD (Rnd)	346.7 (315.0, 477.2)	170.4 (54.35, 288.7)	0.51 (0.14, 0.82)
pcFBD (Div)	350.9 (315.0, 484.9)	239.5 (196.4, 315.6)	0.00 (0.00, 0.02)
$d\sim$ Exp(100)
FBD (Rnd)	861.8 (315.0, 1483.)	832.2 (25.4, 1436.2)	0.00 (0.0, 0.009)
FBD (Div)	321.7 (315.0, 343.1)	265.3 (223.2, 315.6)	0.00 (0.00, 0.02)
pcFBD (Rnd)	355.9 (315.0, 494.3)	209.3 (61.69, 322.4)	0.40 (0.07, 0.78)
pcFBD (Div)	358.8 (315.0, 517.0)	233.5 (202.7, 252.0)	0.00 (0.00, 0.03)
$d\sim$ Exp(10000)
FBD (Rnd)	326.5 (315.0, 365.5)	279.2 (64.86, 345.0)	0.00 (0.00, 0.01)
FBD (Div)	320.0 (315.0, 335.7)	235.8 (196.0, 286.0)	0.00 (0.00, 0.02)
pcFBD (Rnd)	570.2 (315.0, 1098.)	458.3 (235.0, 1025.)	0.00 (0.00, 0.01)
pcFBD (Div)	360.4 (315.0, 519.6)	246.0 (219.9, 252.0)	0.00 (0.00, 0.01)

Notes: Numbers represent the median and the 95% HPD interval. The hyperprior for $d=\lambda -\mu$ is Exponential (rate parameter specified in the table), for $r=\mu /\lambda$ and $s=\psi /(\mu +\psi )$ is Beta(1, 1), for the root age is Exponential with mean 396 and offset 315, and for the fossils are uniform on the time interval of the stratum in which they were found (Table 3). For the full empirical analysis (Table 5), the intermediate prior on $d$ was used.