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�%070.?�,�;,=,80?0=�?:�.3,920�,..:=/492�?:�4?’>�;=:;:>,7�;=:-,-474?D��02� The MCMC sampler will use the following moves: With prob. Chain will change 2.86 % param. 1 (tratio) with Dirichlet proposal 2.86 % param. 2 (revmat) with Dirichlet proposal 2.86 % param. 3 (revmat) with Dirichlet proposal 2.86 % param. 4 (state frequencies) with Dirichlet proposal 2.86 % param. 5 (state frequencies) with Dirichlet proposal 2.86 % param. 6 (state frequencies) with Dirichlet proposal 2.86 % param. 7 (state frequencies) with Dirichlet proposal 2.86 % param. 8 (state frequencies) with Dirichlet proposal 2.86 % param. 9 (gamma shape) with multiplier 2.86 % param. 10 (gamma shape) with multiplier 2.86 % param. 11 (gamma shape) with multiplier 2.86 % param. 12 (gamma shape) with multiplier 2.86 % param. 13 (gamma shape) with multiplier 2.86 % param. 14 (prop. invar. sites) with beta proposal 2.86 % param. 15 (rate multiplier) with Dirichlet proposal 14.29 % param. 16 (topology and branch lengths) with LOCAL 42.86 % param. 16 (topology and branch lengths) with extending TBR

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TreeSetViz: http://comet.lehman.cuny.edu/treeviz/ Mesquite: http://www.mesquiteproject.org/mesquite/mesquite.html

Hillis et al., Analysis and Visualization of Tree Space, Syst. Biol., 54(3): 471-482.

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(ii) �..0;?,9.0�=,?0>�:1�;,=,80?0=�@;/,?0>

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(i) �:=8�:1�?30�?480�>0=40>�;7:?>�:1�;,=,80?0=�0>?48,?0>

�.:9?49@:@>�;,=,80?0=>��02��>@->?4?@?4:9�=,?0>���&=,.0=

�/4>.=0?0�;,=,80?0=>�

B,=8�,9/�1@EED�.,?0=;477,=>

�/4>?,9.0>�,8:92�>,8;70/�?:;:7:240>��&=00%0?'4E

Page 38: >512 :@>;0A/@5;:@; 5-3:;?5:3 #1>2;>9-:/1treethinkers.org/wp-content/uploads/2013/03/MCMC.pdf · ' !>18-d10/8;/7 95?/$ %?,?4>?4.,7 >?48,?4:9:1"3d7:209d 9!@?7490 $-@1?;2&>-5@ b;8a@5;:

�>>0>>492������"0=1:=8,9.0��4,29:>?4.>��,>0/�:9�%49270�$@9>

Acceptance rates for the moves in the "cold" chain of run 1: With prob. Chain accepted changes to 13.61 % param. 1 (revmat) with Dirichlet proposal

.

.

. 0.04 % param. 34 (rate multiplier) Dirichlet proposal 6.59 % param. 35 (topology and branch lengths) TBR 14.06 % param. 35 (topology and branch lengths) LOCAL

Acceptance rates for the moves in the "cold" chain of run 1: With prob. Chain accepted changes to 33.30 % param. 1 (revmat) with Dirichlet proposal

.

.

. 19.13 % param. 34 (rate multiplier) Dirichlet proposal 17.40 % param. 35 (topology and branch lengths) TBR 29.76 % param. 35 (topology and branch lengths) LOCAL

-,/�84C492 -0??0=�84C492

�C,8;70��&=,.0=�;7:?>�:1�=07,?4A0�=,?0�8@7?4;740=>�1=:8�?B:��=�,D0>�=@9>

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�>>0>>492������"0=1:=8,9.0��4,29:>?4.>��,>0/�:9�%49270�$@9>

��4C492�/4,29:>?4.>

�,..0;?,9.0�=,?0>�.,9�-0�.:9?=:770/�-D�A,=D492�?30�>.,70�:1�?30���?@9492�;,=,80?0=>�1:=�?30�=070A,9?�;=:;:>,7�80.3,94>8>

?:�49.=0,>0�=,?0>��/0.=0,>0�>.,70���A4.0�A0=>,

(ii) �..0;?,9.0�=,?0>�:1�;,=,80?0=�@;/,?0>

�.:9?49@:@>���/4>.=0?0�;,=,80?0=>���=�,D0>�����%&��0?.=,?0>�>3:@7/�4/0,77D�1,77�49�?30�~20-70%�=,920

(i) �:=8�:1�?30�?480�>0=40>�;7:?>�:1�;,=,80?0=�0>?48,?0>

�.:9?49@:@>�;,=,80?0=>��02��>@->?4?@?4:9�=,?0>���&=,.0=

�/4>.=0?0�;,=,80?0=>�

B,=8�,9/�1@EED�.,?0=;477,=>

�/4>?,9.0>�,8:92�>,8;70/�?:;:7:240>��&=00%0?'4E

Page 40: >512 :@>;0A/@5;:@; 5-3:;?5:3 #1>2;>9-:/1treethinkers.org/wp-content/uploads/2013/03/MCMC.pdf · ' !>18-d10/8;/7 95?/$ %?,?4>?4.,7 >?48,?4:9:1"3d7:209d 9!@?7490 $-@1?;2&>-5@ b;8a@5;:

�>>0>>492������"0=1:=8,9.0��4,29:>?4.>��,>0/�:9�%49270�$@9>

��4C492�/4,29:>?4.>

(iii) �:=8�:1�?30�8,=249,7�;:>?0=4:=�;=:-,-474?D�/09>4?40>

�.:9?49@:@>�;,=,80?0=>��02��>@->?4?@?4:9�=,?0>���&=,.0=

-0B,=0�:1�;:=.@;490�=:,/6477

�,..0;?,9.0�=,?0>�.,9�-0�.:9?=:770/�-D�A,=D492�?30�>.,70�:1�?30���?@9492�;,=,80?0=>�1:=�?30�=070A,9?�;=:;:>,7�80.3,94>8>

?:�49.=0,>0�=,?0>��/0.=0,>0�>.,70���A4.0�A0=>,

(ii) �..0;?,9.0�=,?0>�:1�;,=,80?0=�@;/,?0>

�.:9?49@:@>���/4>.=0?0�;,=,80?0=>���=�,D0>�����%&��0?.=,?0>�>3:@7/�4/0,77D�1,77�49�?30�~20-70%�=,920

(i) �:=8�:1�?30�?480�>0=40>�;7:?>�:1�;,=,80?0=�0>?48,?0>

�.:9?49@:@>�;,=,80?0=>��02��>@->?4?@?4:9�=,?0>���&=,.0=

�/4>.=0?0�;,=,80?0=>�

B,=8�,9/�1@EED�.,?0=;477,=>

�/4>?,9.0>�,8:92�>,8;70/�?:;:7:240>��&=00%0?'4E

Page 41: >512 :@>;0A/@5;:@; 5-3:;?5:3 #1>2;>9-:/1treethinkers.org/wp-content/uploads/2013/03/MCMC.pdf · ' !>18-d10/8;/7 95?/$ %?,?4>?4.,7 >?48,?4:9:1"3d7:209d 9!@?7490 $-@1?;2&>-5@ b;8a@5;:

�>>0>>492������"0=1:=8,9.0��4,29:>?4.>��,>0/�:9�%49270�$@9>

Acceptance rates for the moves in the "cold" chain of run 1: With prob. Chain accepted changes to 13.61 % param. 1 (revmat) with Dirichlet proposal

.

.

. 0.04 % param. 34 (rate multiplier) Dirichlet proposal 6.59 % param. 35 (topology and branch lengths) TBR 14.06 % param. 35 (topology and branch lengths) LOCAL

Acceptance rates for the moves in the "cold" chain of run 1: With prob. Chain accepted changes to 33.30 % param. 1 (revmat) with Dirichlet proposal

.

.

. 19.13 % param. 34 (rate multiplier) Dirichlet proposal 17.40 % param. 35 (topology and branch lengths) TBR 29.76 % param. 35 (topology and branch lengths) LOCAL

Frequency

m{2}0.2 0.25 0.3 0.35 0.4 0.450

100

200

300

400

500

600

700

800

Frequency

meanRate2E-4 3E-4 4E-4 5E-4 6E-4 7E-4 8E-4 9E-4 1E-30

100

200

300

400

500

-,/�84C492

�C,8;70��",=,80?0=�0>?48,?0>�1:=�=07,?4A0�=,?0�8@7?4;740=>�1=:8�?B:��=�,D0>�=@9>

-0??0=�84C492

Page 42: >512 :@>;0A/@5;:@; 5-3:;?5:3 #1>2;>9-:/1treethinkers.org/wp-content/uploads/2013/03/MCMC.pdf · ' !>18-d10/8;/7 95?/$ %?,?4>?4.,7 >?48,?4:9:1"3d7:209d 9!@?7490 $-@1?;2&>-5@ b;8a@5;:

�>>0>>492������"0=1:=8,9.0��4,29:>?4.>��,>0/�:9�%49270�$@9>

Acceptance rates for the moves in the "cold" chain of run 1: With prob. Chain accepted changes to 13.61 % param. 1 (revmat) with Dirichlet proposal

.

.

. 0.04 % param. 34 (rate multiplier) Dirichlet proposal 6.59 % param. 35 (topology and branch lengths) TBR 14.06 % param. 35 (topology and branch lengths) LOCAL

Acceptance rates for the moves in the "cold" chain of run 1: With prob. Chain accepted changes to 33.30 % param. 1 (revmat) with Dirichlet proposal

.

.

. 19.13 % param. 34 (rate multiplier) Dirichlet proposal 17.40 % param. 35 (topology and branch lengths) TBR 29.76 % param. 35 (topology and branch lengths) LOCAL

Frequency

m{2}0.2 0.25 0.3 0.35 0.4 0.450

100

200

300

400

500

600

700

800

Frequency

meanRate2E-4 3E-4 4E-4 5E-4 6E-4 7E-4 8E-4 9E-4 1E-30

100

200

300

400

500

-,/�84C492

�C,8;70��",=,80?0=�0>?48,?0>�1:=�=07,?4A0�=,?0�8@7?4;740=>�1=:8�?B:��=�,D0>�=@9>

-0??0=�84C492

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(iv)��A@;/;>>18-@5;:�@591����&��;2�<->-91@1>�?-9<81?

⇢k =

Pn�ki=1 (xi � x̄)(xi+k � x̄)Pn

i=1(xi � x̄)2

)1�C;A80�1D<1/@�@41�kth�8-3�-A@;/;>>18-@5;:�@;�.1�?9-881>�-?�k�5:/>1-?1?��;A>��?@�-:0��@4�0>-C?�?4;A80�.1�81??�/;>>18-@10�@4-:�;A>��?@�-:0��:0�0>-C?�

�2�-A@;/;>>18-@5;:�5?�?@588�>18-@5B18E�4534�2;>�45341>�B-8A1?�;2�k��@45?�5:05/-@1?�4534�013>11�;2�/;>>18-@5;:�.1@C11:�;A>�0>-C?�-:0�?8;C�95D5:3

�>>0>>492������"0=1:=8,9.0��4,29:>?4.>��,>0/�:9�%49270�$@9>

��4C492�/4,29:>?4.>

�&41�8-3�k�-A@;/;>>18-@5;:�+>4;,7�5?�@41�/;>>18-@5;:�1B1>E�0>-C�-:0�5@?�kth�8-3�

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0 5 10 15 20 25 30

0.0

0.2

0.4

0.6

0.8

1.0

Lag

ACF

Good Mixing

0 10 20 30 40

0.0

0.2

0.4

0.6

0.8

1.0

Lag

ACF

Bad Mixing

(iv)��A@;/;>>18-@5;:�@591����&��;2�<->-91@1>�?-9<81?

�>>0>>492������"0=1:=8,9.0��4,29:>?4.>��,>0/�:9�%49270�$@9>

��4C492�/4,29:>?4.>

-0??0=�84C492 -,/�84C492

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��%,8;70�>4E0�/4,29:>?4.>

(i) �110.?4A0�%,8;70�%4E0���%%��/4,29:>?4.

�9@8-0=�:1�>,8;70>,@?:.:==07,?4:9�?480����&�

�.:9?49@:@>�;,=,80?0=>��02��>@->?4?@?4:9�=,?0>���&=,.0=

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�>>0>>492������"0=1:=8,9.0��4,29:>?4.>��,>0/�:9�%49270�$@9>

�C,8;70���%%�A,7@0>�1:=�=07,?4A0�=,?0�8@7?4;740=>�1=:8�?B:��=�,D0>�=@9>

7:B�49?09>4?D

Page 47: >512 :@>;0A/@5;:@; 5-3:;?5:3 #1>2;>9-:/1treethinkers.org/wp-content/uploads/2013/03/MCMC.pdf · ' !>18-d10/8;/7 95?/$ %?,?4>?4.,7 >?48,?4:9:1"3d7:209d 9!@?7490 $-@1?;2&>-5@ b;8a@5;:

�>>0>>492������"0=1:=8,9.0��4,29:>?4.>��,>0/�:9�%49270�$@9>

��%,8;70�>4E0�/4,29:>?4.>

(i) �110.?4A0�%,8;70�%4E0���%%��/4,29:>?4.

(ii) �:=8�:1�?30�8,=249,7�;:>?0=4:=�;=:-,-474?D�/09>4?40>

�.:9?49@:@>�;,=,80?0=>��02��>@->?4?@?4:9�=,?0>���&=,.0=

-=:?30=�:1�;:=.@;490�=:,/6477

�9@8-0=�:1�>,8;70>,@?:.:==07,?4:9�?480����&�

�.:9?49@:@>�;,=,80?0=>��02��>@->?4?@?4:9�=,?0>���&=,.0=

09>@=0�%���.:8;74,9.0�

Page 48: >512 :@>;0A/@5;:@; 5-3:;?5:3 #1>2;>9-:/1treethinkers.org/wp-content/uploads/2013/03/MCMC.pdf · ' !>18-d10/8;/7 95?/$ %?,?4>?4.,7 >?48,?4:9:1"3d7:209d 9!@?7490 $-@1?;2&>-5@ b;8a@5;:

�>>0>>492������"0=1:=8,9.0��4,29:>?4.>��,>0/�:9�%49270�$@9>

�C,8;70��",=,80?0=�0>?48,?0>�1:=�80,9�=,?0�8@7?4;740=>�1=:8����%&�=@9>

7:B�49?09>4?D

Frequency

meanRate2E-4 3E-4 4E-4 5E-4 6E-4 7E-4 8E-40

2.5

5

7.5

10

12.5

���.D.70>

-0??0=�49?09>4?D

Frequency

meanRate2E-4 3E-4 4E-4 5E-4 6E-4 7E-4 8E-4 9E-40

10

20

30

40

50

60

70

���.D.70>

�49,/0<@,?0�.3,49�7092?3

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�>>0>>492������"0=1:=8,9.0��4,29:>?4.>��,>0/�:9�%49270�$@9>

�C,8;70��",=,80?0=�0>?48,?0>�1:=�80,9�=,?0�8@7?4;740=>�1=:8����%&�=@9>

7:B�49?09>4?D

Frequency

meanRate2E-4 3E-4 4E-4 5E-4 6E-4 7E-4 8E-40

2.5

5

7.5

10

12.5

���.D.70>

-0??0=�49?09>4?D

Frequency

meanRate2E-4 3E-4 4E-4 5E-4 6E-4 7E-4 8E-4 9E-40

25

50

75

100

125

����.D.70>

�49,/0<@,?0�.3,49�7092?3

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�>>0>>492������"0=1:=8,9.0��4,29:>?4.>��,>0/�:9�%49270�$@9>

Frequency

m{2}0.2 0.25 0.3 0.35 0.4 0.450

100

200

300

400

500

600

700

800

Frequency

meanRate2E-4 3E-4 4E-4 5E-4 6E-4 7E-4 8E-4 9E-4 1E-30

100

200

300

400

500

�C,8;70��",=,80?0=�0>?48,?0>�1:=�=07,?4A0�=,?0�8@7?4;740=>�1=:8�?B:��=�,D0>�=@9>

7:B�49?09>4?D -0??0=�49?09>4?D

���.D.70> ����.D.70>

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�>>0>>492������"0=1:=8,9.0��4,29:>?4.>��,>0/�:9�%49270�$@9>

alpha p

�����;,?3:7:240>

",=,80?0=�49?0=,.?4:9�-0?B009�I+G�84C?@=0�1:=�,8:92�>4?0�=,?0�A,=4,?4:9

Frequency

mtDNA2.alpha0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.60

100

200

300

400

500

Frequency

mtDNA2.pInv0.1 0.15 0.2 0.25 0.3 0.35 0.40

100

200

300

400

500

Page 52: >512 :@>;0A/@5;:@; 5-3:;?5:3 #1>2;>9-:/1treethinkers.org/wp-content/uploads/2013/03/MCMC.pdf · ' !>18-d10/8;/7 95?/$ %?,?4>?4.,7 >?48,?4:9:1"3d7:209d 9!@?7490 $-@1?;2&>-5@ b;8a@5;:

�>>0>>492������"0=1:=8,9.0��4,29:>?4.>��,>0/�:9�%49270�$@9>

�8@7?4�8:/,7�8,=249,7�/09>4?40>�>?08�1=:8�9:9�4/09?414,-474?D�

�@>0�G B4?3�,//4?4:9,7�/4>.=0?0�=,?0�.,?02:=40>�

alpha p

�����;,?3:7:240>

",=,80?0=�49?0=,.?4:9�-0?B009�I+G�84C?@=0�1:=�,8:92�>4?0�=,?0�A,=4,?4:9

Frequency

mtDNA2.alpha0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.60

100

200

300

400

500

Frequency

mtDNA2.pInv0.1 0.15 0.2 0.25 0.3 0.35 0.40

100

200

300

400

500

Page 53: >512 :@>;0A/@5;:@; 5-3:;?5:3 #1>2;>9-:/1treethinkers.org/wp-content/uploads/2013/03/MCMC.pdf · ' !>18-d10/8;/7 95?/$ %?,?4>?4.,7 >?48,?4:9:1"3d7:209d 9!@?7490 $-@1?;2&>-5@ b;8a@5;:

�;;=:C48,?492�?30��:49?�":>?0=4:=�"=:-,-474?D��09>4?D�B4?3��0?=:;:74>��:@;70>�����

$;.;@�%=A-0>;:��

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�;;=:C48,?492�?30��:49?�":>?0=4:=�"=:-,-474?D��09>4?D�B4?3��0?=:;:74>��:@;70>�����

$;.;@�%=A-0>;:��

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�!���������������������������������������������$�������� ������������������ �������

���������������#�����!���������������������������������$�������$��� �����������������������$�����������������$����� �

N����������������������������������������������������������������������������������$�������$�

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��������������������������������������������������$�������$��� �����������������������������$��������!�����������������������������$�������� �����

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1

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2

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1000

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1. Run m ≥ 2 chains of length 2n from overdispersed starting values.

3. Calculate the within-chain and between-chain variance.

4. Calculate the estimated variance of the parameter as a weighted sum of the within-chain and between-chain variance.

2. Discard the first n draws of each chain.

5. Calculate the PSRF.

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95% Cred. Interval ---------------------- Parameter Mean Variance Lower Upper Median PSRF * ------------------------------------------------------------------------------------------- TL{all} 4.921609 2.998138 2.836000 7.295000 5.056000 9.084 kappa{4,5} 3.095696 0.054125 2.667623 3.587024 3.085271 1.000 alpha{5} 1.006544 0.087721 0.606472 1.738482 0.950093 1.000 pinvar{1} 0.307396 0.009357 0.095913 0.471070 0.316173 1.000 m{1} 0.264226 0.009315 0.146502 0.421870 0.244468 5.507 m{2} 0.040919 0.000227 0.022205 0.065884 0.037425 5.279 m{3} 2.721453 7.157157 0.039001 5.544253 5.030560 69.564 m{4} 2.125810 3.568002 0.199137 4.044249 3.917338 150.012 m{5} 0.188768 0.004373 0.109303 0.295129 0.170624 5.749 -------------------------------------------------------------------------------------------

95% Cred. Interval ---------------------- Parameter Mean Variance Lower Upper Median PSRF * ------------------------------------------------------------------------------------------- TL{all} 0.073893 0.000034 0.063000 0.086000 0.074000 1.000 kappa{2,3} 3.236308 0.366904 2.199024 4.587719 3.190195 1.000 m{1} 1.285838 0.028345 0.980634 1.630387 1.278161 1.000 m{2} 1.423906 0.015507 1.182596 1.664627 1.423610 1.000 m{3} 0.589346 0.005341 0.453175 0.736459 0.587617 1.001 -------------------------------------------------------------------------------------------

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You can never be absolutely certain that the MCMC is reliable, you can only identify when something has gone wrong. Gelman

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You can never be absolutely certain that the MCMC is reliable, you can only identify when something has gone wrong. Gelman

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