1. The +I+G Models
…an aside

2. Modelling Rate Variation
Not every site in a sequence evolves at the same rate

3. Assign sites to rate categories
Gamma distribution (+G model) 1
Rate categories come from a discrete approximation of the Gamma probability distribution
Invariant sites (+I model) 2
Rates are either “No Change” or “Some Change”
1: Yang (1993)
2: Hasegawa, Kishino and Yano (1987)

4. Using both +I and +G Done by Gu et al. in 1995
Estimate a proportion, p0 , of invariable sites
Fit the remaining sites to a gamma distribution

5. Criticism
“This model is somewhat pathological as the gamma distribution with α ≤ 1 already allows for sites with very low rates; […] adding a proportion of invariable sites creates a strong correlation between p0 and α, making it impossible to estimate both parameters reliably” Ziheng Yang, Computational Molecular Evolution (2006)

6. Identifiability
Sullivan et al. (1999)

7. Theoretical Justification
PROOF!
(2001)

8. Theoretical Justification
DISPROOF!
(2001)
(2008)

9. Theoretical Justification
PROOF! (again)
(2001)
(2011)

10. Theoretical Justification
PROOF! (again)
(2001)
(2011)
+I+G is proved to work under a continuous Gamma distribution
All implementations use discrete approximations to the Gamma
It is not clear if +I+G is identifiable under the approximation