Dataset	DS1	DS2	DS3	DS4	DS5	DS6	DS7	DS8
Reference	[8]	[6]	[37]	[9]	[17]	[43]	[38]	[30]
#Taxa (N)	27	29	36	41	50	50	59	64
#Sites (M)	1949	2520	1812	1137	378	1133	1824	1008
MrBayes SS	-7108.42	-26367.57	-33735.44	-13330.06	-8214.51	-6724.07	-37332.76	-8649.88
[29, 42]	(0.18)	(0.48)	(0.5)	(0.54)	(0.28)	(0.86)	(2.42)	(1.75)
(Replication)	-7107.81	-26366.45	-33732.79	-13328.40	-8209.17	-6721.54	-37331.85	-8646.18
(0.25)	(0.40)	(0.63)	(0.48)	(0.46)	(0.77)	(3.08)	(1.19)
VBPI-GNN	-7108.41	-26367.73	-33735.12	-13329.94	-8214.64	-6724.37	-37332.04	-8650.65
[40]	(0.14)	(0.07)	(0.09)	(0.19)	(0.38)	(0.4)	(0.26)	(0.45)
CSMC	-8306.76	-27884.37	-35381.01	-15019.21	-8940.62	-8029.51	-11013.57
[34, 16]	(166.27)	(226.6)	(218.18)	(100.61)	(46.44)	(83.67)	(113.49)
VCSMC	-9180.34	-28700.7	-37211.2	-17106.1	-9449.65	-9296.66
[24, 16]	(170.27)	(4892.67)	(397.97)	(362.74)	(2578.58)	(2046.7)
φ-CSMC	-7290.36	-30568.49	-33798.06	-13582.24	-8367.51	-7013.83	-9209.18
[16]	(7.23)	(31.34)	(6.62)	(35.08)	(8.87)	(16.99)	(18.03)
N(diag,2)	-7126.79	-26440.54	-33814.98	-13356.21	-8251.99	-6747.15	-37526.41	-8727.93
(10.06)	(28.78)	(20.31)	(9.48)	(9.43)	(15.21)	(66.28)	(43.33)
LOO(3)	-7130.60	-26375.10	-33737.71	-13345.55	-8236.99	-6747.46	-37375.93	-8716.61
(10.66)	(11.75)	(2.32)	(3.26)	(6.29)	(6.90)	(28.86)	(26.32)
LOO(3)+	-7128.36	-26369.93	-33735.91	-13346.03	-8236.13	-6751.97	-37430.82	-8691.38
(9.77)	(0.25)	(0.13)	(4.54)	(5.71)	(11.24)	(67.19)	(10.70)
N(diag,4)	-7123.95	-26382.91	-33762.45	-13341.62	-8241.07	-6735.78	-37396.04	-8679.48
(12.03)	(16.89)	(10.68)	(3.64)	(6.56)	(5.25)	(26.39)	(27.55)
LOO(3)	-7120.88	-26368.53	-33736.04	-13338.99	-8238.16	-6735.59	-37357.86	-8665.54
(13.15)	(0.05)	(0.09)	(6.08)	(0.52)	(4.51)	(10.76)	(5.66)
LOO(3)+	-7119.81	-26368.49	-33735.92	-13339.79	-8236.69	-6736.74	-37353.08	-8665.99
(11.71)	(0.10)	(0.15)	(4.55)	(4.70)	(3.55)	(16.97)	(7.53)
N(full,2)	-7129.70	-26487.71	-33807.05	-13353.30	-8251.01	-6750.00	-37487.49	-8736.81
(6.14)	(54.79)	(22.97)	(5.92)	(10.34)	(11.91)	(50.43)	(52.38)
LOO(3)	-7132.35	-26391.00	-33736.98	-13347.17	-8237.75	-6752.46	-37462.07	-8684.38
(6.89)	(11.88)	(1.89)	(7.77)	(6.08)	(8.64)	(54.40)	(7.98)
LOO(3)+	-7122.76	-26380.59	-33736.93	-13343.21	-8239.96	-6753.84	-37419.02	-8691.96
(10.81)	(14.39)	(2.10)	(2.14)	(4.84)	(14.30)	(35.94)	(13.51)
N(full,4)	-7120.03	-26378.55	-33753.20	-13342.27	-8237.33	-6734.51	-37373.32	-8662.53
(11.92)	(11.05)	(3.03)	(2.71)	(5.41)	(1.95)	(10.08)	(4.58)
LOO(3)	-7124.62	-26368.49	-33736.03	-13337.74	-8234.18	-6734.49	-37347.46	-8666.63
(12.33)	(0.13)	(0.16)	(1.71)	(6.11)	(3.14)	(11.93)	(7.86)
LOO(3)+	-7123.37	-26368.51	-33735.99	-13337.06	-8241.25	-6734.63	-37352.30	-8666.39
(11.28)	(0.09)	(0.05)	(1.45)	(8.15)	(2.18)	(12.32)	(7.54)
WNV(full,2)	-7126.89	-26444.84	-33823.74	-13358.16	-8251.45	-6745.60	-37516.88	-8719.44
(10.06)	(27.91)	(15.62)	(9.79)	(9.72)	(8.36)	(69.88)	(60.54)
LOO(3)	-7130.67	-26380.41	-33737.75	-13346.94	-8239.36	-6741.63	-37382.28	-8690.41
(10.67)	(14.40)	(2.48)	(4.25)	(4.62)	(3.23)	(31.96)	(15.92)
LOO(3)+	-7128.40	-26375.28	-33736.91	-13347.32	-8235.41	-6742.40	-37411.28	-8683.22
(9.78)	(11.78)	(1.91)	(4.42)	(5.70)	(1.94)	(56.74)	(13.13)
WNV(full,4)	-7122.10	-26381.84	-33759.19	-13342.81	-8243.92	-6733.38	-37369.36	-8666.85
(12.29)	(17.18)	(9.95)	(3.45)	(6.74)	(0.79)	(13.45)	(10.63)
LOO(3)	-7120.94	-26368.52	-33735.98	-13339.77	-8236.42	-6735.12	-37341.92	-8673.15
(13.11)	(0.03)	(0.08)	(3.84)	(3.63)	(2.52)	(9.15)	(0.97)
LOO(3)+	-7125.78	-26368.51	-33736.00	-13342.38	-8235.03	-6736.20	-37345.80	-8666.68
(13.10)	(0.10)	(0.18)	(6.35)	(5.36)	(1.91)	(11.13)	(5.78)
WNV(full,2)	-7124.63	-26458.50	-33804.63	-13358.16	-8251.09	-6748.78	-37484.98	-8717.27
(8.09)	(32.42)	(20.76)	(1.20)	(7.46)	(7.38)	(34.91)	(28.49)
LOO(3)	-7125.94	-26391.02	-33736.98	-13344.13	-8236.90	-6753.86	-37416.00	-8684.90
(13.07)	(11.89)	(1.96)	(0.22)	(5.13)	(10.68)	(3.12)	(12.81)
LOO(3)+	-7115.19	-26385.21	-33736.97	-13343.95	-8239.55	-6747.61	-37431.76	-8683.54
(8.16)	(13.77)	(1.93)	(1.76)	(4.72)	(6.87)	(43.65)	(3.57)
WNV(full,4)	-7111.55	-26379.48	-33757.79	-13342.71	-8240.87	-6735.14	-37377.86	-8663.51
(0.07)	(11.60)	(8.07)	(1.61)	(9.80)	(2.64)	(29.48)	(6.85)
LOO(3)	-7119.77	-26368.44	-33736.01	-13339.26	-8234.06	-6733.91	-37350.77	-8671.32
(11.80)	(0.13)	(0.03)	(3.19)	(7.53)	(0.57)	(11.74)	(5.99)
LOO(3)+	-7116.09	-26368.54	-33735.85	-13337.42	-8233.89	-6735.90	-37358.96	-8660.48
(10.67)	(0.12)	(0.12)	(1.32)	(6.63)	(1.13)	(13.06)	(0.78)