User:Carl Boettiger/Notebook/Comparative Phylogenetics/2010/06/09
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Graham & Peter Meeting
Fantastic meeting with Graham and Peter today, covered a lot of ground.
I briefly sketched the multivariate normal solution for joint probability across the tree under the regimes model. The original regimes approach did not take advantage of the fact that the solution to the joint probability across the tree is multivariate normal given the painting. This allows the calculation to be partitioned as outlined in Saturday's entry:
is a likelihood ratio and is referred to as the weighting function. The last equality in the above equation motivates the estimator
This is the importance sampling estimator of and is unbiased. That is, the estimation procedure is to generate i.i.d. samples from and for each sample which exceeds , the estimate is incremented by the weight evaluated at the sample value. The results are averaged over trials.
Partitioning the problem
and proposing paintings directly, we can MCMC over the space of possible paintings , OU parameters and transition matrices . Still, as this problem is hard in the discrete case over (BayesTraits), optimizing the MCMC will still be interesting...
Wainwright Lab Meeting, 4-6pm
Presented three potential questions to focus on for the Evolution talk:
The group clearly indicated that I should focus on the third, AIC topic, as it is the furthest along and the most immediate impact to the audience. Surprising to me as it is also the least biologically driven. Back to the drawing board now to figure out how to tell this story clearly and succinctly.