Bayesian reversible-jump MCMC: deflating inflated support values

Recently, on the Botanical Society of America Student ListServ, commonly-used phylogenetic methods were being discussed, and a specific problem was raised with regard to Bayesian phylogenetic inference: it has been documented that, within a Bayesian framework, support values (in the form of posterior probabilities) can become inflated or skewed.  This is an issue of special interest to me, since I commonly use Bayesian inference alongside other phylogenetic methods in my research (e.g., Hodkinson & Lendemer 2010, Hodkinson & Lutzoni 2009, Miadlikowska et al. 2006).  I decided to contribute to the discussion by sending the following message to the ListServ:

"
I have found that Bayesian phylogenetic inference methods can do amazing things (like finding the same topology with one gene that only emerges with two or three genes in an analysis based on maximum parsimony (MP) or maximum likelihood (ML)).  However, this problem of Bayesian methods inflating support (especially at short internodes) makes me feel very suspicious of any given Bayesian posterior probability value. Therefore, if I want to evaluate support, I always look at MP- and/or ML-bootstrap proportions.  The Bayesian problem that we're talking about has been discussed by Lewis et al. (2005), and they state that the solution is to use reversible-jump MCMC (rjMCMC).  However, I have wondered myself how I could actually implement this.

I recently found an article that clearly stated the fact that MrBayes and BEAST cannot perform rjMCMC (Kodandaramaiah et al. 2010): http://dx.doi.org/10.1016/j.ympev.2009.08.012
In this article, they used Phycas for their rjMCMC analyses.

So I went to the Phycas manual (Lewis et al. 2010), and found that it does give the nitty-gritty of how to implement this (see section 2.3 on 'Polytomy Priors'):
http://phylo.bio.ku.edu/phycas/manual.pdf
I have not yet done it myself, but I am very excited to try it out!  Perhaps now I will have more faith in my posterior probabilities!
"

The Phycas manual gives the best, most concise summary of the issue that I have seen anywhere:

"
A solution to the 'Star Tree Paradox' problem was proposed by Lewis, Holder, and Holsinger (2005). Their solution was to use reversible-jump MCMC to allow unresolved tree topologies to be sampled during the course of a Bayesian phylogenetic analysis in addition to fully-resolved tree topologies. If the time between speciation events is so short (or the substitution rate so low) that no substitutions occurred along a particular internal edge in the true tree, then use of the polytomy prior proposed by Lewis, Holder, and Holsinger (2005) can improve inference by giving the Bayesian model a 'way out.' That is, it is not required to find a fully resolved tree, but is allowed to place a lot of posterior probability mass on a less-than-fully-resolved topology. Please refer to the Lewis, Holder, and Holsinger (2005) paper for details.
"

Please post comments here if you have performed these types of analyses and have any additional insights!

-Brendan

P.S. Please see my more recent posts for the details of using Phycas:
http://squamules.blogspot.com/2011/06/installing-and-running-phycas.html
http://squamules.blogspot.com/2011/06/writing-phycas-script.html

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Works Cited:

Hodkinson, B. P., and J. C. Lendemer. 2011. Molecular analyses reveal semi-cryptic species in Xanthoparmelia tasmanica. Bibliotheca Lichenologica 106: 115-126.
Download draft (PDF file)
Download alignment (NEXUS file)

Hodkinson, B. P., and F. Lutzoni. 2009. A microbiotic survey of lichen-associated bacteria reveals a new lineage from the Rhizobiales. Symbiosis 49: 163-180.
Download publication (PDF file)
Download alignment (NEXUS file)

Kodandaramaiah U., C. Pena, M. F. Braby, R. Grund, C. J. Muller, S. Nylin, and N. Wahlberg. 2010. Phylogenetics of Coenonymphina (Nymphalidae: Satyrinae) and the problem of rooting rapid radiations. Molecular Phylogenetics and Evolution 54(2): 386-394.
View Publication (webpage)

Lewis, P. O., M. T. Holder, and K. E. Holsinger. 2005. Polytomies and Bayesian phylogenetic inference. Systematic Biology 54(2): 241-253
View Publication (webpage)

Lewis, P. O., M. T. Holder, and D. L. Swofford. 2010. Phycas User Manual, Version 1.2.0.
View Manual (PDF file)

Miadlikowska, J., F. Kauff, V. Hofstetter, E. Fraker, M. Grube, J. Hafellner, V. Reeb, B. P. Hodkinson, M. Kukwa, R. Lücking, G. Hestmark, M. Garcia Otalora, A. Rauhut, B. Büdel, C. Scheidegger, E. Timdal, S. Stenroos, I. Brodo, G. Perlmutter, D. Ertz, P. Diederich, J. C. Lendemer, P. May, C. L. Schoch, A. E. Arnold, C. Gueidan, E. Tripp, R. Yahr, C. Robertson, and F. Lutzoni. 2006. New insights into classification and evolution of the Lecanoromycetes (Pezizomycotina, Ascomycota) from phylogenetic analyses of three ribosomal RNA- and two protein-coding genes. Mycologia 98: 1088-1103.
Download publication (PDF file)
Download supplement (PDF file)
Download alignment (zipped NEXUS file)

A recipe with fungi and algae!

I have never actually eaten a lichen, but I was thinking the other day about how I do like to eat both fungi and algae, especially together.  Several years ago, my wife and I took a Korean cooking class, where we learned some great recipes.  The one that we enjoy cooking the most is Bee-bim Bop, a dish with all kinds of veggies served over rice.  We usually serve it like a salad bar, where people can put whatever items they want on their rice then mix it themselves.  In addition to the veggies, both fungi (mushrooms) and algae (seaweed) are a traditional part of the dish, and help to form a flavor symbiosis of sorts.

Here's our recipe for Bee-bim Bop:

Ingredients (you can vary the quantities of each vegetable depending on your preference)
Rice (cook ~1/2 cup per person)
Shiitake mushrooms
Bean Sprouts (~1 cup)
Eggs (to be whipped, fried, and sliced; we use 3 eggs)
Carrots (to be cut into matchsticks and fried; 1 big carrot should do it)
Onion (to be chopped and fried)
Cucumber (to be cut into matchsticks)
Lettuce (1 small head or less)
Tofu (firm, cubed; ~ 1 cup)
green onion tops for garnish
Seasonings: salt, garlic, sesame oil, sesame seeds, sugar, soy sauce, and cayenne pepper

Now there is a lot of vegetable chopping to be done!

Directions:
1. Soak dried mushrooms at least 30 minutes
2. Wash and get rice cooking.
3. Prepare marinade sauce for mushrooms and tofu: 1t salt, 1/2t garlic, and 1t sesame oil (make as much as you need using those proportions).  Put the tofu (raw) in a bowl with some of the sauce to marinate while you do the next few steps.
3. Wash the bean sprouts and put them in a pot with a little water (not quite covering them) and a pinch of salt. Cook them, covered, about 15 minutes on medium-high heat, then drain and place in a bowl.
4. Heat some sesame oil in a frying pan for the eggs. Mix the eggs and a dash of salt vigorously with a fork and then pour them into the hot pan. Let it spread out as much as possible. Flip it after a couple minutes, and when the "pancake" is cooked through, take it out of the frying pan and slice it into long, thin strips.
5. Cut the carrots and onions into thin pieces and fry them (with salt) in the hot oil until the carrots are tender.
6. Drain the re-hydrated mushrooms and squeeze excess water out of them.  Mix them with some of the sauce from step 3 and fry.
7. Fry the tofu in hot oil with black pepper.
8. Drain the cooked bean sprouts and season them with a little bit of salt, garlic, sesame seeds, and sesame oil.
9. Cut the cucumber and lettuce into long thin strips. 
10. Prepare Goo Choo Chang sauce: mix 1/2t cayenne pepper, 1/2t garlic, 1t sesame seeds, 1T soy sauce,  1T sugar, and 1T sesame oil.  You'll want to put this in a little container for pouring at the dinner table.

When you have it all ready, you should have on the table a big bowl of rice and small bowls with the mushrooms, sprouts, egg strips, carrots & onions, tofu, cucumber, and lettuce. 

To Serve:
Put cooked rice into individual serving bowls.
Put vegetables, eggs, tofu, and mushrooms on top of the rice.
Pour Go Choo Chang sauce over bowl.
Mix thoroughly, and add more sesame oil to the mixture depending on your preference.
Serve with Kim Chee and Laver seaweed (thin and crispy... not sushi nori).
 

The ingredients on display (with a small one who is quite excited about the algae)!

The final product, with algae around the edges and fungi in the mix with the rest.

All that's left at this point is to give your hands a thorough washing and dig in; we typically eat this by using the little sheets of laver as our utensils!

Enjoy!

-Brendan