Bayesian II

Spring 2010

Major Issues in Phylogenetic BI

• Have we reached convergence?• If so, do we have a large enough sample of the

posterior?

Have we reached convergence?

• Look at the trace plots of the posterior probability (not reliable)

• Convergence Diagnostics:– Average Standard Deviation of the Split

frequencies: compares the node frequencies between the independent runs (the closer to zero, the better, but it is not clear how small it should be)

– Potential Scale Reduction Factor (PSRF): the closer to one, the better

Convergence Diagnostics

• By default performs two independent analyses starting from different random trees (mcmc nruns=2)

• Average standard deviation of clade frequencies calculated and presented during the run (mcmc mcmcdiagn=yes diagnfreq=1000) and written to file (.mcmc)

• Standard deviation of each clade frequency and potential scale reduction for branch lengths calculated with sumt

• Potential scale reduction calculated for all substitution model parameters with sump

Have we reached convergence?PSRF (sump command)

• Model parameter summaries over all 3 runs sampled in files• "Ligia16SCOI28SGulfnoADLGGTRGbygene.nex.run1.p", "Ligia16SCOI28SGulfnoADLGGTRGbygene.nex.run2.p" etc:• (Summaries are based on a total of 162003 samples from 3 runs)• (Each run produced 60001 samples of which 54001 samples were included)•  • 95% Cred. Interval• ----------------------• Parameter Mean Variance Lower Upper Median PSRF *• -------------------------------------------------------------------------------------------• TL{all} 1.578535 0.008255 1.417000 1.771000 1.573000 1.000• r(A<->C){1} 0.036558 0.000133 0.016961 0.061987 0.035555 1.000• r(A<->G){1} 0.405681 0.002305 0.314283 0.502156 0.404811 1.000• r(A<->T){1} 0.044360 0.000120 0.025482 0.068209 0.043489 1.000• r(C<->G){1} 0.022006 0.000120 0.004949 0.047283 0.020532 1.000• r(C<->T){1} 0.472371 0.002269 0.379396 0.565910 0.472279 1.000• r(G<->T){1} 0.019024 0.000079 0.005074 0.039579 0.017873 1.000• pi(A){1} 0.292047 0.000390 0.254211 0.331575 0.291819 1.000• pi(C){1} 0.204875 0.000278 0.173364 0.238514 0.204491 1.000• pi(G){1} 0.214907 0.000324 0.180784 0.251150 0.214522 1.000• pi(T){1} 0.288171 0.000367 0.251355 0.326301 0.287933 1.000• alpha{1} 0.183087 0.000419 0.146653 0.226929 0.181792 1.000• -------------------------------------------------------------------------------------------• * Convergence diagnostic (PSRF = Potential scale reduction factor [Gelman• and Rubin, 1992], uncorrected) should approach 1 as runs converge. The• values may be unreliable if you have a small number of samples. PSRF should• only be used as a rough guide to convergence since all the assumptions• that allow one to interpret it as a scale reduction factor are not met in• the phylogenetic context.•  

Copyright restrictions may apply.

Nylander, J. A.A. et al. Bioinformatics 2008 24:581-583; doi:10.1093/bioinformatics/btm388

Are We There Yet (AWTY)?

Empirical Data: two independent runs 300,000,000 generations: complex model with three partitions (by codon): the bad news

Plotted in Tracer

Empirical Data: two independent runs 300,000,000 generations: complex model with three partitions (by codon): the good news; splits were highly correlated between the

two runs

Plotted in AWTY: http://ceb.scs.fsu.edu/awty

Empirical Data: two independent runs 300,000,000 generations: complex model with three partitions (by codon): the good news; splits were highly correlated between the

two runs

Plotted in AWTY: http://ceb.scs.fsu.edu/awty

What caused the difference in posterior probabilities? Estimation of particular parameters

Yes we have reached convergence:Do we have a large enough sample of the posterior?

• Long runs are better than short one, but how long?• Good mixing: “Examine the acceptance rates of the

proposal mechanisms used in your• analysis (output at the end of the run)• The Metropolis proposals used by MrBayes work

best when their acceptance rate is neither too low nor too high. A rough guide is to try to get them within the range of 10 % to 70 %”

Acceptance Rates Analysis used 2373953.05 seconds of CPU time on processor 0 Likelihood of best state for "cold" chain of run 1 was -9688.70 Likelihood of best state for "cold" chain of run 2 was -9865.21 Likelihood of best state for "cold" chain of run 3 was -9887.59 Likelihood of best state for "cold" chain of run 4 was -9895.35 Acceptance rates for the moves in the "cold" chain of run 1: With prob. Chain accepted changes to 42.86 % param. 1 (revmat) with Dirichlet proposal 21.42 % param. 2 (revmat) with Dirichlet proposal 55.92 % param. 3 (revmat) with Dirichlet proposal 29.18 % param. 4 (state frequencies) with Dirichlet proposal 12.22 % param. 5 (state frequencies) with Dirichlet proposal 24.61 % param. 6 (state frequencies) with Dirichlet proposal 41.02 % param. 7 (gamma shape) with multiplier 31.74 % param. 8 (gamma shape) with multiplier 79.95 % param. 9 (gamma shape) with multiplier 40.16 % param. 10 (rate multiplier) with Dirichlet proposal 15.80 % param. 11 (topology and branch lengths) with extending TBR 23.62 % param. 11 (topology and branch lengths) with LOCAL

tree 1 tree 2 tree 3

)|( Xf

Posterior probability distribution

Parameter space

Post

erio

r pro

babi

lity

cold chain

heated chain

Metropolis-coupled Markov chain Monte Carlo

a. k. a.

MCMCMC

a. k. a.

(MC)3

cold chain

hot chain

cold chain

hot chain

cold chain

hot chain

unsuccessful swap

cold chain

hot chain

cold chain

hot chain

cold chain

hot chain

cold chain

hot chain

successful swap

cold chain

hot chain

cold chain

hot chain

cold chain

hot chain

successful swap

cold chain

hot chain

Improving Convergence

(Only if convergence diagnostics indicate problem!)• Change tuning parameters of proposals to bring

acceptance rate into the range 10 % to 70 %• Propose changes to ‘difficult’ parameters more often• Use different proposal mechanisms• Change heating temperature to bring acceptance rate

of swaps between adjacent chains into the range 10 % to 70 %.

• Run the chain longer• Increase the number of heated chains• Make the model more realistic

Sam

pled

val

ueTarget distribution

Too modest proposalsAcceptance rate too highPoor mixing

Too bold proposalsAcceptance rate too lowPoor mixing

Moderately bold proposalsAcceptance rate intermediateGood mixing