Question

The Effect Of The "Invgamma" Setting On Computing The Site Likelihood In Mrbayes?

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14.0 years ago

Douglas S. Stones ▴ 280

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.

phylogenetics • 4.5k views

ADD COMMENT • link updated 14.0 years ago by Botond Sipos ★ 1.7k • written 14.0 years ago by Douglas S. Stones ▴ 280

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This might be an implementation detail that the developers might know more about.

ADD REPLY • link 14.0 years ago by Istvan Albert 102k

score 4 · Answer 1 · 2011-01-09

Based solely on theory I think the answer to your question is the following:

Let $p_{inv}$ be the proportion of invariant sites, n the number of gamma categories and $r_{i}$ the rate corresponding to the i^th category of the discrete gamma model (as determined by the actual gamma shape parameter). Let alt text be the root-node conditional likelihood calculated with a given fixed rate $r_{i}$ (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:

$L(p_{inv}, r_{1}, r_{2},\ldots,r_{n}) = p_{inv} L(0) + (1 - p_{inv}) [\frac{1}{n} \sum_{i=1}^{n} \L(r_i)]$

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 $p_{inv}$ parameter is the weight of the likelihood calculated with the fixed rate zero.