The "invgamma" setting in MrBayes sets a gamma-distributed rate variation across sites and a proportion of invariable sites. The gamma distribution is approximated by several discrete rate categories.
According to Felsenstein 1981 (Evolutionary Trees from DNA Sequences: A Maximum Likelihood Approach), the site likelihood is computed from the root-node conditional likelihoods by $L=\sum_{s_0} \pi_{s_0} L_{s_0}^{(0)}$. However, when the invgamma setting is used in MrBayes, there will be a root-node conditional likelihood for each discrete rate category.
What effect do the discrete rates have on computing the site likelihood from the root-node conditional likelihoods?
What effect does the "proportion of invariable sites" have on the computation?
In particular, I'm looking for some documentation that answers these questions in detail.
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written
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Douglas S. Stones •
1 answer
Based solely on theory I think the answer to your question is the following:
Let 
be the proportion of invariant sites, n the number of gamma categories and 
the rate corresponding to the 
category of the discrete gamma model (as determined by the actual gamma shape parameter). Let 
be the root-node conditional likelihood calculated with a given fixed rate (let's just drop the notation for the site and the model parameters).
The +I+G model assumes that the sites rates are distributed according to mixture of two discrete distributions:
- the fixed rate zero with a weight p_{inv}
- the discrete gamma distribution with a weight 1 - p_{inv}
The full likelihood is calculated as:
![L(p_{inv}, r_{1}, r_{2},\ldots,r_{n}) = p_{inv} L(0) + (1 - p_{inv}) [\frac{1}{n} \sum_{i=1}^{n} \L(r_i)]](http://mathurl.com/2u4tv9m.png)
The likelihood computation must be performed multiple times on the same site with different fixed rates, so using a +I+G model slows down the computation n+1-fold compared to the fixed rate model.
The parameter is the weight of the likelihood calculated with the fixed rate zero.
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This might be an implementation detail that the developers might know more about.