8.3: Using Maximum likelihood to Estimate Parameters of the Mk model
The algorithm in the appendix below gives the likelihood for any particular discrete-state Markov model on a tree, but requires us to specify a value of the rate parameter q . In the example given, this rate parameter q = 1.0 corresponds to a lnL of -6.5. But is this the best value of q to use for our Mk model? Probably not. We can use maximum likelihood to find a better estimate of this parameter.
If we apply the pruning algorithm across a range of different values of q , the likelihood changes. To find the ML estimate of q , we can again use numerical optimization methods, calculating the likelihood by pruning for many values of q and finding the maximum.
Applying this method to the lizard data, we obtain a maximum liklihood estimate of q = 0.001850204 corresponding to l n L = −80.487176.
The example above considers maximization of a single parameter, which is a relatively simple problem. When we extend this to a multi-parameter model – for example, the extended Mk model will all rates different (ARD) – maximizing the likelihood becomes much more difficult. R packages solve this problem by using sophisticated algorithms and applying them multiple times to make sure that the value found is actually a maximum.